Number fields¶

We define a quartic number field and its quadratic extension:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<y> = NumberField(x^4 - 420*x^2 + 40000)
sage: z = y^5/11; z
420/11*y^3 - 40000/11*y
sage: R.<y> = PolynomialRing(K)
sage: f = y^2 + y + 1
sage: L.<a> = K.extension(f); L
Number Field in a with defining polynomial y^2 + y + 1 over its base field
sage: KL.<b> = NumberField([x^4 - 420*x^2 + 40000, x^2 + x + 1]); KL
Number Field in b0 with defining polynomial x^4 - 420*x^2 + 40000 over its base field

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(4) - Integer(420)*x**Integer(2) + Integer(40000), names=('y',)); (y,) = K._first_ngens(1)
>>> z = y**Integer(5)/Integer(11); z
420/11*y^3 - 40000/11*y
>>> R = PolynomialRing(K, names=('y',)); (y,) = R._first_ngens(1)
>>> f = y**Integer(2) + y + Integer(1)
>>> L = K.extension(f, names=('a',)); (a,) = L._first_ngens(1); L
Number Field in a with defining polynomial y^2 + y + 1 over its base field
>>> KL = NumberField([x**Integer(4) - Integer(420)*x**Integer(2) + Integer(40000), x**Integer(2) + x + Integer(1)], names=('b',)); (b,) = KL._first_ngens(1); KL
Number Field in b0 with defining polynomial x^4 - 420*x^2 + 40000 over its base field

We do some arithmetic in a tower of relative number fields:

Sage

sage: K.<cuberoot2> = NumberField(x^3 - 2)
sage: L.<cuberoot3> = K.extension(x^3 - 3)
sage: S.<sqrt2> = L.extension(x^2 - 2)
sage: S
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
sage: sqrt2 * cuberoot3
cuberoot3*sqrt2
sage: (sqrt2 + cuberoot3)^5
(20*cuberoot3^2 + 15*cuberoot3 + 4)*sqrt2 + 3*cuberoot3^2 + 20*cuberoot3 + 60
sage: cuberoot2 + cuberoot3
cuberoot3 + cuberoot2
sage: cuberoot2 + cuberoot3 + sqrt2
sqrt2 + cuberoot3 + cuberoot2
sage: (cuberoot2 + cuberoot3 + sqrt2)^2
(2*cuberoot3 + 2*cuberoot2)*sqrt2 + cuberoot3^2 + 2*cuberoot2*cuberoot3 + cuberoot2^2 + 2
sage: cuberoot2 + sqrt2
sqrt2 + cuberoot2
sage: a = S(cuberoot2); a
cuberoot2
sage: a.parent()
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) - Integer(2), names=('cuberoot2',)); (cuberoot2,) = K._first_ngens(1)
>>> L = K.extension(x**Integer(3) - Integer(3), names=('cuberoot3',)); (cuberoot3,) = L._first_ngens(1)
>>> S = L.extension(x**Integer(2) - Integer(2), names=('sqrt2',)); (sqrt2,) = S._first_ngens(1)
>>> S
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
>>> sqrt2 * cuberoot3
cuberoot3*sqrt2
>>> (sqrt2 + cuberoot3)**Integer(5)
(20*cuberoot3^2 + 15*cuberoot3 + 4)*sqrt2 + 3*cuberoot3^2 + 20*cuberoot3 + 60
>>> cuberoot2 + cuberoot3
cuberoot3 + cuberoot2
>>> cuberoot2 + cuberoot3 + sqrt2
sqrt2 + cuberoot3 + cuberoot2
>>> (cuberoot2 + cuberoot3 + sqrt2)**Integer(2)
(2*cuberoot3 + 2*cuberoot2)*sqrt2 + cuberoot3^2 + 2*cuberoot2*cuberoot3 + cuberoot2^2 + 2
>>> cuberoot2 + sqrt2
sqrt2 + cuberoot2
>>> a = S(cuberoot2); a
cuberoot2
>>> a.parent()
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field

Warning

Doing arithmetic in towers of relative fields that depends on canonical coercions is currently VERY SLOW. It is much better to explicitly coerce all elements into a common field, then do arithmetic with them there (which is quite fast).

AUTHORS:

William Stein (2004, 2005): initial version
Steven Sivek (2006-05-12): added support for relative extensions
William Stein (2007-09-04): major rewrite and documentation
Robert Bradshaw (2008-10): specified embeddings into ambient fields
Simon King (2010-05): improved coercion from GAP
Jeroen Demeyer (2010-07, 2011-04): upgraded PARI (Issue #9343, Issue #10430, Issue #11130)
Robert Harron (2012-08): added is_CM(), complex_conjugation(), and maximal_totally_real_subfield()
Christian Stump (2012-11): added conversion to universal cyclotomic field
Julian Rueth (2014-04-03): absolute number fields are unique parents
Vincent Delecroix (2015-02): comparisons/floor/ceil using embeddings
Kiran Kedlaya (2016-05): relative number fields hash based on relative polynomials
Peter Bruin (2016-06): made number fields fully satisfy unique representation
John Jones (2017-07): improved check for is_galois(), add is_abelian(), building on work in patch by Chris Wuthrich
Anna Haensch (2018-03): added quadratic_defect()
Michael Daub, Chris Wuthrich (2020-09-01): added Dirichlet characters for abelian fields

class sage.rings.number_field.number_field.CyclotomicFieldFactory[source]¶

Bases: UniqueFactory

Return the \(n\)-th cyclotomic field, where \(n\) is a positive integer, or the universal cyclotomic field if \(n=0\).

For the documentation of the universal cyclotomic field, see UniversalCyclotomicField.

INPUT:

n – nonnegative integer (default: \(0\))
names – name of generator (default: zetan)
bracket – defines the brackets in the case of \(n=0\), and is ignored otherwise. Can be any even length string, with '()' being the default.
embedding – boolean or \(n\)-th root of unity in an ambient field (default: True)

EXAMPLES:

If called without a parameter, we get the universal cyclotomic field:

Sage

sage: CyclotomicField()                                                         # needs sage.libs.gap
Universal Cyclotomic Field

Python

>>> from sage.all import *
>>> CyclotomicField()                                                         # needs sage.libs.gap
Universal Cyclotomic Field

We create the \(7\)th cyclotomic field \(\QQ(\zeta_7)\) with the default generator name.

Sage

sage: k = CyclotomicField(7); k
Cyclotomic Field of order 7 and degree 6
sage: k.gen()
zeta7

Python

>>> from sage.all import *
>>> k = CyclotomicField(Integer(7)); k
Cyclotomic Field of order 7 and degree 6
>>> k.gen()
zeta7

The default embedding sends the generator to the complex primitive \(n\)-th root of unity of least argument.

Sage

sage: CC(k.gen())
0.623489801858734 + 0.781831482468030*I

Python

>>> from sage.all import *
>>> CC(k.gen())
0.623489801858734 + 0.781831482468030*I

Cyclotomic fields are of a special type.

Sage

sage: type(k)
<class 'sage.rings.number_field.number_field.NumberField_cyclotomic_with_category'>

Python

>>> from sage.all import *
>>> type(k)
<class 'sage.rings.number_field.number_field.NumberField_cyclotomic_with_category'>

We can specify a different generator name as follows.

Sage

sage: k.<z7> = CyclotomicField(7); k
Cyclotomic Field of order 7 and degree 6
sage: k.gen()
z7

Python

>>> from sage.all import *
>>> k = CyclotomicField(Integer(7), names=('z7',)); (z7,) = k._first_ngens(1); k
Cyclotomic Field of order 7 and degree 6
>>> k.gen()
z7

The \(n\) must be an integer.

Sage

sage: CyclotomicField(3/2)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(3)/Integer(2))
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer

The degree must be nonnegative.

Sage

sage: CyclotomicField(-1)
Traceback (most recent call last):
...
ValueError: n (=-1) must be a positive integer

Python

>>> from sage.all import *
>>> CyclotomicField(-Integer(1))
Traceback (most recent call last):
...
ValueError: n (=-1) must be a positive integer

The special case \(n=1\) does not return the rational numbers:

Sage

sage: CyclotomicField(1)
Cyclotomic Field of order 1 and degree 1

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(1))
Cyclotomic Field of order 1 and degree 1

Due to their default embedding into \(\CC\), cyclotomic number fields are all compatible.

Sage

sage: cf30 = CyclotomicField(30)
sage: cf5 = CyclotomicField(5)
sage: cf3 = CyclotomicField(3)
sage: cf30.gen() + cf5.gen() + cf3.gen()
zeta30^6 + zeta30^5 + zeta30 - 1
sage: cf6 = CyclotomicField(6) ; z6 = cf6.0
sage: cf3 = CyclotomicField(3) ; z3 = cf3.0
sage: cf3(z6)
zeta3 + 1
sage: cf6(z3)
zeta6 - 1
sage: cf9 = CyclotomicField(9) ; z9 = cf9.0
sage: cf18 = CyclotomicField(18) ; z18 = cf18.0
sage: cf18(z9)
zeta18^2
sage: cf9(z18)
-zeta9^5
sage: cf18(z3)
zeta18^3 - 1
sage: cf18(z6)
zeta18^3
sage: cf18(z6)**2
zeta18^3 - 1
sage: cf9(z3)
zeta9^3

Python

>>> from sage.all import *
>>> cf30 = CyclotomicField(Integer(30))
>>> cf5 = CyclotomicField(Integer(5))
>>> cf3 = CyclotomicField(Integer(3))
>>> cf30.gen() + cf5.gen() + cf3.gen()
zeta30^6 + zeta30^5 + zeta30 - 1
>>> cf6 = CyclotomicField(Integer(6)) ; z6 = cf6.gen(0)
>>> cf3 = CyclotomicField(Integer(3)) ; z3 = cf3.gen(0)
>>> cf3(z6)
zeta3 + 1
>>> cf6(z3)
zeta6 - 1
>>> cf9 = CyclotomicField(Integer(9)) ; z9 = cf9.gen(0)
>>> cf18 = CyclotomicField(Integer(18)) ; z18 = cf18.gen(0)
>>> cf18(z9)
zeta18^2
>>> cf9(z18)
-zeta9^5
>>> cf18(z3)
zeta18^3 - 1
>>> cf18(z6)
zeta18^3
>>> cf18(z6)**Integer(2)
zeta18^3 - 1
>>> cf9(z3)
zeta9^3

create_key(n=0, names=None, embedding=True)[source]¶: Create the unique key for the cyclotomic field specified by the parameters.

create_object(version, key, **extra_args)[source]¶: Create the unique cyclotomic field defined by key.

sage.rings.number_field.number_field.GaussianField()[source]¶: The field \(\QQ[i]\).

sage.rings.number_field.number_field.NumberField(polynomial, name, check=None, names=True, embedding=None, latex_name=None, assume_disc_small=None, maximize_at_primes=False, structure=None, latex_names=None, **kwds)[source]¶

Return the number field (or tower of number fields) defined by the irreducible polynomial.

INPUT:

polynomial – a polynomial over \(\QQ\) or a number field, or a list of such polynomials
names (or name) – string or list of strings, the names of the generators
check – boolean (default: True); do type checking and irreducibility checking
embedding – None, an element, or a list of elements, the images of the generators in an ambient field (default: None)
latex_names (or latex_name) – None, a string, or a list of strings (default: None); how the generators are printed for latex output
assume_disc_small – boolean (default: False); if True, assume that no square of a prime greater than PARI’s primelimit (which should be 500000). Only applies for absolute fields at present.
maximize_at_primes – None or a list of primes (default: None); if not None, then the maximal order is computed by maximizing only at the primes in this list, which completely avoids having to factor the discriminant, but of course can lead to wrong results; only applies for absolute fields at present.
structure – None, a list or an instance of structure.NumberFieldStructure (default: None), internally used to pass in additional structural information, e.g., about the field from which this field is created as a subfield.

We accept implementation and prec attributes for compatibility with AlgebraicExtensionFunctor but we ignore them as they are not used.

EXAMPLES:

Sage

sage: z = QQ['z'].0
sage: K = NumberField(z^2 - 2, 's'); K
Number Field in s with defining polynomial z^2 - 2
sage: s = K.0; s
s
sage: s*s
2
sage: s^2
2

Python

>>> from sage.all import *
>>> z = QQ['z'].gen(0)
>>> K = NumberField(z**Integer(2) - Integer(2), 's'); K
Number Field in s with defining polynomial z^2 - 2
>>> s = K.gen(0); s
s
>>> s*s
2
>>> s**Integer(2)
2

Constructing a relative number field:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^2 - 2)
sage: R.<t> = K[]
sage: L.<b> = K.extension(t^3 + t + a); L
Number Field in b with defining polynomial t^3 + t + a over its base field
sage: L.absolute_field('c')
Number Field in c with defining polynomial x^6 + 2*x^4 + x^2 - 2
sage: a*b
a*b
sage: L(a)
a
sage: L.lift_to_base(b^3 + b)
-a

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> L = K.extension(t**Integer(3) + t + a, names=('b',)); (b,) = L._first_ngens(1); L
Number Field in b with defining polynomial t^3 + t + a over its base field
>>> L.absolute_field('c')
Number Field in c with defining polynomial x^6 + 2*x^4 + x^2 - 2
>>> a*b
a*b
>>> L(a)
a
>>> L.lift_to_base(b**Integer(3) + b)
-a

Constructing another number field:

Sage

sage: k.<i> = NumberField(x^2 + 1)
sage: R.<z> = k[]
sage: m.<j> = NumberField(z^3 + i*z + 3)
sage: m
Number Field in j with defining polynomial z^3 + i*z + 3 over its base field

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = k._first_ngens(1)
>>> R = k['z']; (z,) = R._first_ngens(1)
>>> m = NumberField(z**Integer(3) + i*z + Integer(3), names=('j',)); (j,) = m._first_ngens(1)
>>> m
Number Field in j with defining polynomial z^3 + i*z + 3 over its base field

Number fields are globally unique:

Sage

sage: K.<a> = NumberField(x^3 - 5)
sage: a^3
5
sage: L.<a> = NumberField(x^3 - 5)
sage: K is L
True

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) - Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> a**Integer(3)
5
>>> L = NumberField(x**Integer(3) - Integer(5), names=('a',)); (a,) = L._first_ngens(1)
>>> K is L
True

Equality of number fields depends on the variable name of the defining polynomial:

Sage

sage: x = polygen(QQ, 'x'); y = polygen(QQ, 'y')
sage: k.<a> = NumberField(x^2 + 3)
sage: m.<a> = NumberField(y^2 + 3)
sage: k
Number Field in a with defining polynomial x^2 + 3
sage: m
Number Field in a with defining polynomial y^2 + 3
sage: k == m
False

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x'); y = polygen(QQ, 'y')
>>> k = NumberField(x**Integer(2) + Integer(3), names=('a',)); (a,) = k._first_ngens(1)
>>> m = NumberField(y**Integer(2) + Integer(3), names=('a',)); (a,) = m._first_ngens(1)
>>> k
Number Field in a with defining polynomial x^2 + 3
>>> m
Number Field in a with defining polynomial y^2 + 3
>>> k == m
False

In case of conflict of the generator name with the name given by the preparser, the name given by the preparser takes precedence:

Sage

sage: K.<b> = NumberField(x^2 + 5, 'a'); K
Number Field in b with defining polynomial x^2 + 5

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(5), 'a', names=('b',)); (b,) = K._first_ngens(1); K
Number Field in b with defining polynomial x^2 + 5

One can also define number fields with specified embeddings, may be used for arithmetic and deduce relations with other number fields which would not be valid for an abstract number field.

Sage

sage: K.<a> = NumberField(x^3 - 2, embedding=1.2)
sage: RR.coerce_map_from(K)
Composite map:
  From: Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?
  To:   Real Field with 53 bits of precision
  Defn:   Generic morphism:
          From: Number Field in a with defining polynomial x^3 - 2
                with a = 1.259921049894873?
          To:   Real Lazy Field
          Defn: a -> 1.259921049894873?
        then
          Conversion via _mpfr_ method map:
          From: Real Lazy Field
          To:   Real Field with 53 bits of precision
sage: RR(a)
1.25992104989487
sage: 1.1 + a
2.35992104989487
sage: b = 1/(a+1); b
1/3*a^2 - 1/3*a + 1/3
sage: RR(b)
0.442493334024442
sage: L.<b> = NumberField(x^6 - 2, embedding=1.1)
sage: L(a)
b^2
sage: a + b
b^2 + b

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) - Integer(2), embedding=RealNumber('1.2'), names=('a',)); (a,) = K._first_ngens(1)
>>> RR.coerce_map_from(K)
Composite map:
  From: Number Field in a with defining polynomial x^3 - 2 with a = 1.259921049894873?
  To:   Real Field with 53 bits of precision
  Defn:   Generic morphism:
          From: Number Field in a with defining polynomial x^3 - 2
                with a = 1.259921049894873?
          To:   Real Lazy Field
          Defn: a -> 1.259921049894873?
        then
          Conversion via _mpfr_ method map:
          From: Real Lazy Field
          To:   Real Field with 53 bits of precision
>>> RR(a)
1.25992104989487
>>> RealNumber('1.1') + a
2.35992104989487
>>> b = Integer(1)/(a+Integer(1)); b
1/3*a^2 - 1/3*a + 1/3
>>> RR(b)
0.442493334024442
>>> L = NumberField(x**Integer(6) - Integer(2), embedding=RealNumber('1.1'), names=('b',)); (b,) = L._first_ngens(1)
>>> L(a)
b^2
>>> a + b
b^2 + b

Note that the image only needs to be specified to enough precision to distinguish roots, and is exactly computed to any needed precision:

Sage

sage: RealField(200)(a)
1.2599210498948731647672106072782283505702514647015079800820

Python

>>> from sage.all import *
>>> RealField(Integer(200))(a)
1.2599210498948731647672106072782283505702514647015079800820

One can embed into any other field:

Sage

sage: K.<a> = NumberField(x^3 - 2, embedding=CC.gen() - 0.6)
sage: CC(a)
-0.629960524947436 + 1.09112363597172*I

sage: # needs sage.rings.padics
sage: L = Qp(5)
sage: f = polygen(L)^3 - 2
sage: K.<a> = NumberField(x^3 - 2, embedding=f.roots()[0][0])
sage: a + L(1)
4 + 2*5^2 + 2*5^3 + 3*5^4 + 5^5 + 4*5^6 + 2*5^8 + 3*5^9 + 4*5^12
 + 4*5^14 + 4*5^15 + 3*5^16 + 5^17 + 5^18 + 2*5^19 + O(5^20)
sage: L.<b> = NumberField(x^6 - x^2 + 1/10, embedding=1)
sage: K.<a> = NumberField(x^3 - x + 1/10, embedding=b^2)
sage: a + b
b^2 + b
sage: CC(a) == CC(b)^2
True
sage: K.coerce_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - x + 1/10 with a = b^2
  To:   Number Field in b with defining polynomial x^6 - x^2 + 1/10
        with b = 0.9724449978911874?
  Defn: a -> b^2

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) - Integer(2), embedding=CC.gen() - RealNumber('0.6'), names=('a',)); (a,) = K._first_ngens(1)
>>> CC(a)
-0.629960524947436 + 1.09112363597172*I

>>> # needs sage.rings.padics
>>> L = Qp(Integer(5))
>>> f = polygen(L)**Integer(3) - Integer(2)
>>> K = NumberField(x**Integer(3) - Integer(2), embedding=f.roots()[Integer(0)][Integer(0)], names=('a',)); (a,) = K._first_ngens(1)
>>> a + L(Integer(1))
4 + 2*5^2 + 2*5^3 + 3*5^4 + 5^5 + 4*5^6 + 2*5^8 + 3*5^9 + 4*5^12
 + 4*5^14 + 4*5^15 + 3*5^16 + 5^17 + 5^18 + 2*5^19 + O(5^20)
>>> L = NumberField(x**Integer(6) - x**Integer(2) + Integer(1)/Integer(10), embedding=Integer(1), names=('b',)); (b,) = L._first_ngens(1)
>>> K = NumberField(x**Integer(3) - x + Integer(1)/Integer(10), embedding=b**Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> a + b
b^2 + b
>>> CC(a) == CC(b)**Integer(2)
True
>>> K.coerce_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - x + 1/10 with a = b^2
  To:   Number Field in b with defining polynomial x^6 - x^2 + 1/10
        with b = 0.9724449978911874?
  Defn: a -> b^2

The QuadraticField and CyclotomicField constructors create an embedding by default unless otherwise specified:

Sage

sage: K.<zeta> = CyclotomicField(15)
sage: CC(zeta)
0.913545457642601 + 0.406736643075800*I
sage: L.<sqrtn3> = QuadraticField(-3)
sage: K(sqrtn3)
2*zeta^5 + 1
sage: sqrtn3 + zeta
2*zeta^5 + zeta + 1

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(15), names=('zeta',)); (zeta,) = K._first_ngens(1)
>>> CC(zeta)
0.913545457642601 + 0.406736643075800*I
>>> L = QuadraticField(-Integer(3), names=('sqrtn3',)); (sqrtn3,) = L._first_ngens(1)
>>> K(sqrtn3)
2*zeta^5 + 1
>>> sqrtn3 + zeta
2*zeta^5 + zeta + 1

Comparison depends on the (real) embedding specified (or the one selected by default). Note that the codomain of the embedding must be QQbar or AA for this to work (see Issue #20184):

Sage

sage: N.<g> = NumberField(x^3 + 2, embedding=1)
sage: 1 < g
False
sage: g > 1
False
sage: RR(g)
-1.25992104989487

Python

>>> from sage.all import *
>>> N = NumberField(x**Integer(3) + Integer(2), embedding=Integer(1), names=('g',)); (g,) = N._first_ngens(1)
>>> Integer(1) < g
False
>>> g > Integer(1)
False
>>> RR(g)
-1.25992104989487

If no embedding is specified or is complex, the comparison is not returning something meaningful.:

Sage

sage: N.<g> = NumberField(x^3 + 2)
sage: 1 < g
False
sage: g > 1
True

Python

>>> from sage.all import *
>>> N = NumberField(x**Integer(3) + Integer(2), names=('g',)); (g,) = N._first_ngens(1)
>>> Integer(1) < g
False
>>> g > Integer(1)
True

Since SageMath 6.9, number fields may be defined by polynomials that are not necessarily integral or monic. The only notable practical point is that in the PARI interface, a monic integral polynomial defining the same number field is computed and used:

Sage

sage: K.<a> = NumberField(2*x^3 + x + 1)
sage: K.pari_polynomial()
x^3 - x^2 - 2

Python

>>> from sage.all import *
>>> K = NumberField(Integer(2)*x**Integer(3) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.pari_polynomial()
x^3 - x^2 - 2

Elements and ideals may be converted to and from PARI as follows:

Sage

sage: pari(a)
Mod(-1/2*y^2 + 1/2*y, y^3 - y^2 - 2)
sage: K(pari(a))
a
sage: I = K.ideal(a); I
Fractional ideal (a)
sage: I.pari_hnf()
[1, 0, 0; 0, 1, 0; 0, 0, 1/2]
sage: K.ideal(I.pari_hnf())
Fractional ideal (a)

Python

>>> from sage.all import *
>>> pari(a)
Mod(-1/2*y^2 + 1/2*y, y^3 - y^2 - 2)
>>> K(pari(a))
a
>>> I = K.ideal(a); I
Fractional ideal (a)
>>> I.pari_hnf()
[1, 0, 0; 0, 1, 0; 0, 0, 1/2]
>>> K.ideal(I.pari_hnf())
Fractional ideal (a)

Here is an example where the field has non-trivial class group:

Sage

sage: L.<b> = NumberField(3*x^2 - 1/5)
sage: L.pari_polynomial()
x^2 - 15
sage: J = L.primes_above(2)[0]; J
Fractional ideal (2, 15*b + 1)
sage: J.pari_hnf()
[2, 1; 0, 1]
sage: L.ideal(J.pari_hnf())
Fractional ideal (2, 15*b + 1)

Python

>>> from sage.all import *
>>> L = NumberField(Integer(3)*x**Integer(2) - Integer(1)/Integer(5), names=('b',)); (b,) = L._first_ngens(1)
>>> L.pari_polynomial()
x^2 - 15
>>> J = L.primes_above(Integer(2))[Integer(0)]; J
Fractional ideal (2, 15*b + 1)
>>> J.pari_hnf()
[2, 1; 0, 1]
>>> L.ideal(J.pari_hnf())
Fractional ideal (2, 15*b + 1)

An example involving a variable name that defines a function in PARI:

Sage

sage: theta = polygen(QQ, 'theta')
sage: M.<z> = NumberField([theta^3 + 4, theta^2 + 3]); M
Number Field in z0 with defining polynomial theta^3 + 4 over its base field

Python

>>> from sage.all import *
>>> theta = polygen(QQ, 'theta')
>>> M = NumberField([theta**Integer(3) + Integer(4), theta**Integer(2) + Integer(3)], names=('z',)); (z,) = M._first_ngens(1); M
Number Field in z0 with defining polynomial theta^3 + 4 over its base field

class sage.rings.number_field.number_field.NumberFieldFactory[source]¶

Bases: UniqueFactory

Factory for number fields.

This should usually not be called directly, use NumberField() instead.

INPUT:

polynomial – a polynomial over \(\QQ\) or a number field
name – string (default: 'a'); the name of the generator
check – boolean (default: True); do type checking and irreducibility checking
embedding – None or an element, the images of the generator in an ambient field (default: None)
latex_name – None or string (default: None); how the generator is printed for latex output
assume_disc_small – boolean (default: False); if True, assume that no square of a prime greater than PARI’s primelimit (which should be 500000). Only applies for absolute fields at present.
maximize_at_primes – None or a list of primes (default: None); if not None, then the maximal order is computed by maximizing only at the primes in this list, which completely avoids having to factor the discriminant, but of course can lead to wrong results; only applies for absolute fields at present.
structure – None or an instance of structure.NumberFieldStructure (default: None), internally used to pass in additional structural information, e.g., about the field from which this field is created as a subfield.

create_key_and_extra_args(polynomial, name, check, embedding, latex_name, assume_disc_small, maximize_at_primes, structure)[source]¶: Create a unique key for the number field specified by the parameters.

create_object(version, key, check)[source]¶: Create the unique number field defined by key.

sage.rings.number_field.number_field.NumberFieldTower(polynomials, names, check=True, embeddings=None, latex_names=None, assume_disc_small=False, maximize_at_primes=None, structures=None)[source]¶

Create the tower of number fields defined by the polynomials in the list polynomials.

INPUT:

polynomials – list of polynomials. Each entry must be polynomial which is irreducible over the number field generated by the roots of the following entries.
names – list of strings or a string, the names of the generators of the relative number fields. If a single string, then names are generated from that string.
check – boolean (default: True); whether to check that the polynomials are irreducible
embeddings – list of elements or None (default: None); embeddings of the relative number fields in an ambient field
latex_names – list of strings or None (default: None); names used to print the generators for latex output
assume_disc_small – boolean (default: False); if True, assume that no square of a prime greater than PARI’s primelimit (which should be 500000). Only applies for absolute fields at present.
maximize_at_primes – None or a list of primes (default: None); if not None, then the maximal order is computed by maximizing only at the primes in this list, which completely avoids having to factor the discriminant, but of course can lead to wrong results; only applies for absolute fields at present.
structures – None or a list (default: None), internally used to provide additional information about the number field such as the field from which it was created.

OUTPUT:

The relative number field generated by a root of the first entry of polynomials over the relative number field generated by root of the second entry of polynomials … over the number field over which the last entry of polynomials is defined.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a,b,c> = NumberField([x^2 + 1, x^2 + 3, x^2 + 5]); k  # indirect doctest
Number Field in a with defining polynomial x^2 + 1 over its base field
sage: a^2
-1
sage: b^2
-3
sage: c^2
-5
sage: (a+b+c)^2
(2*b + 2*c)*a + 2*c*b - 9

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField([x**Integer(2) + Integer(1), x**Integer(2) + Integer(3), x**Integer(2) + Integer(5)], names=('a', 'b', 'c',)); (a, b, c,) = k._first_ngens(3); k  # indirect doctest
Number Field in a with defining polynomial x^2 + 1 over its base field
>>> a**Integer(2)
-1
>>> b**Integer(2)
-3
>>> c**Integer(2)
-5
>>> (a+b+c)**Integer(2)
(2*b + 2*c)*a + 2*c*b - 9

The Galois group is a product of 3 groups of order 2:

Sage

sage: k.absolute_field(names='c').galois_group()                                # needs sage.groups
Galois group 8T3 (2[x]2[x]2) with order 8 of x^8 + 36*x^6 + 302*x^4 + 564*x^2 + 121

Python

>>> from sage.all import *
>>> k.absolute_field(names='c').galois_group()                                # needs sage.groups
Galois group 8T3 (2[x]2[x]2) with order 8 of x^8 + 36*x^6 + 302*x^4 + 564*x^2 + 121

Repeatedly calling base_field allows us to descend the internally constructed tower of fields:

Sage

sage: k.base_field()
Number Field in b with defining polynomial x^2 + 3 over its base field
sage: k.base_field().base_field()
Number Field in c with defining polynomial x^2 + 5
sage: k.base_field().base_field().base_field()
Rational Field

Python

>>> from sage.all import *
>>> k.base_field()
Number Field in b with defining polynomial x^2 + 3 over its base field
>>> k.base_field().base_field()
Number Field in c with defining polynomial x^2 + 5
>>> k.base_field().base_field().base_field()
Rational Field

In the following example the second polynomial is reducible over the first, so we get an error:

Sage

sage: v = NumberField([x^3 - 2, x^3 - 2], names='a')
Traceback (most recent call last):
...
ValueError: defining polynomial (x^3 - 2) must be irreducible

Python

>>> from sage.all import *
>>> v = NumberField([x**Integer(3) - Integer(2), x**Integer(3) - Integer(2)], names='a')
Traceback (most recent call last):
...
ValueError: defining polynomial (x^3 - 2) must be irreducible

We mix polynomial parent rings:

Sage

sage: k.<y> = QQ[]
sage: m = NumberField([y^3 - 3, x^2 + x + 1, y^3 + 2], 'beta'); m
Number Field in beta0 with defining polynomial y^3 - 3 over its base field
sage: m.base_field ()
Number Field in beta1 with defining polynomial x^2 + x + 1 over its base field

Python

>>> from sage.all import *
>>> k = QQ['y']; (y,) = k._first_ngens(1)
>>> m = NumberField([y**Integer(3) - Integer(3), x**Integer(2) + x + Integer(1), y**Integer(3) + Integer(2)], 'beta'); m
Number Field in beta0 with defining polynomial y^3 - 3 over its base field
>>> m.base_field ()
Number Field in beta1 with defining polynomial x^2 + x + 1 over its base field

A tower of quadratic fields:

Sage

sage: K.<a> = NumberField([x^2 + 3, x^2 + 2, x^2 + 1]); K
Number Field in a0 with defining polynomial x^2 + 3 over its base field
sage: K.base_field()
Number Field in a1 with defining polynomial x^2 + 2 over its base field
sage: K.base_field().base_field()
Number Field in a2 with defining polynomial x^2 + 1

Python

>>> from sage.all import *
>>> K = NumberField([x**Integer(2) + Integer(3), x**Integer(2) + Integer(2), x**Integer(2) + Integer(1)], names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a0 with defining polynomial x^2 + 3 over its base field
>>> K.base_field()
Number Field in a1 with defining polynomial x^2 + 2 over its base field
>>> K.base_field().base_field()
Number Field in a2 with defining polynomial x^2 + 1

LaTeX versions of generator names can be specified either as:

Sage

sage: K = NumberField([x^3 - 2, x^3 - 3, x^3 - 5], names=['a', 'b', 'c'],
....:                 latex_names=[r'\alpha', r'\beta', r'\gamma'])
sage: K.inject_variables(verbose=False)
sage: latex(a + b + c)
\alpha + \beta + \gamma

Python

>>> from sage.all import *
>>> K = NumberField([x**Integer(3) - Integer(2), x**Integer(3) - Integer(3), x**Integer(3) - Integer(5)], names=['a', 'b', 'c'],
...                 latex_names=[r'\alpha', r'\beta', r'\gamma'])
>>> K.inject_variables(verbose=False)
>>> latex(a + b + c)
\alpha + \beta + \gamma

or as:

Sage

sage: K = NumberField([x^3 - 2, x^3 - 3, x^3 - 5], names='a', latex_names=r'\alpha')
sage: K.inject_variables()
Defining a0, a1, a2
sage: latex(a0 + a1 + a2)
\alpha_{0} + \alpha_{1} + \alpha_{2}

Python

>>> from sage.all import *
>>> K = NumberField([x**Integer(3) - Integer(2), x**Integer(3) - Integer(3), x**Integer(3) - Integer(5)], names='a', latex_names=r'\alpha')
>>> K.inject_variables()
Defining a0, a1, a2
>>> latex(a0 + a1 + a2)
\alpha_{0} + \alpha_{1} + \alpha_{2}

A bigger tower of quadratic fields:

Sage

sage: K.<a2,a3,a5,a7> = NumberField([x^2 + p for p in [2,3,5,7]]); K
Number Field in a2 with defining polynomial x^2 + 2 over its base field
sage: a2^2
-2
sage: a3^2
-3
sage: (a2+a3+a5+a7)^3
((6*a5 + 6*a7)*a3 + 6*a7*a5 - 47)*a2 + (6*a7*a5 - 45)*a3 - 41*a5 - 37*a7

Python

>>> from sage.all import *
>>> K = NumberField([x**Integer(2) + p for p in [Integer(2),Integer(3),Integer(5),Integer(7)]], names=('a2', 'a3', 'a5', 'a7',)); (a2, a3, a5, a7,) = K._first_ngens(4); K
Number Field in a2 with defining polynomial x^2 + 2 over its base field
>>> a2**Integer(2)
-2
>>> a3**Integer(2)
-3
>>> (a2+a3+a5+a7)**Integer(3)
((6*a5 + 6*a7)*a3 + 6*a7*a5 - 47)*a2 + (6*a7*a5 - 45)*a3 - 41*a5 - 37*a7

The function can also be called by name:

Sage

sage: NumberFieldTower([x^2 + 1, x^2 + 2], ['a','b'])
Number Field in a with defining polynomial x^2 + 1 over its base field

Python

>>> from sage.all import *
>>> NumberFieldTower([x**Integer(2) + Integer(1), x**Integer(2) + Integer(2)], ['a','b'])
Number Field in a with defining polynomial x^2 + 1 over its base field

class sage.rings.number_field.number_field.NumberField_absolute(polynomial, name, latex_name=None, check=True, embedding=None, assume_disc_small=False, maximize_at_primes=None, structure=None)[source]¶

Bases: NumberField_generic

Function to initialize an absolute number field.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K = NumberField(x^17 + 3, 'a'); K
Number Field in a with defining polynomial x^17 + 3
sage: type(K)
<class 'sage.rings.number_field.number_field.NumberField_absolute_with_category'>
sage: TestSuite(K).run()

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(17) + Integer(3), 'a'); K
Number Field in a with defining polynomial x^17 + 3
>>> type(K)
<class 'sage.rings.number_field.number_field.NumberField_absolute_with_category'>
>>> TestSuite(K).run()

abs_val(v, iota, prec=None)[source]¶

Return the value \(|\iota|_{v}\).

INPUT:

v – a place of K, finite (a fractional ideal) or infinite (element of K.places(prec))
iota – an element of K
prec – (default: None) the precision of the real field

OUTPUT: the absolute value as a real number

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<xi> = NumberField(x^3 - 3)
sage: phi_real = K.places()[0]
sage: phi_complex = K.places()[1]
sage: v_fin = tuple(K.primes_above(3))[0]

sage: K.abs_val(phi_real, xi^2)
2.08008382305190

sage: K.abs_val(phi_complex, xi^2)
4.32674871092223

sage: K.abs_val(v_fin, xi^2)
0.111111111111111

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(3), names=('xi',)); (xi,) = K._first_ngens(1)
>>> phi_real = K.places()[Integer(0)]
>>> phi_complex = K.places()[Integer(1)]
>>> v_fin = tuple(K.primes_above(Integer(3)))[Integer(0)]

>>> K.abs_val(phi_real, xi**Integer(2))
2.08008382305190

>>> K.abs_val(phi_complex, xi**Integer(2))
4.32674871092223

>>> K.abs_val(v_fin, xi**Integer(2))
0.111111111111111

Check that Issue #28345 is fixed:

Sage

sage: K.abs_val(v_fin, K.zero())
0.000000000000000

Python

>>> from sage.all import *
>>> K.abs_val(v_fin, K.zero())
0.000000000000000

absolute_degree()[source]¶

A synonym for degree().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_degree()
2

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.absolute_degree()
2

absolute_different()[source]¶

A synonym for different().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_different()
Fractional ideal (2)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.absolute_different()
Fractional ideal (2)

absolute_discriminant()[source]¶

A synonym for discriminant().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_discriminant()
-4

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.absolute_discriminant()
-4

absolute_generator()[source]¶

An alias for sage.rings.number_field.number_field.NumberField_generic.gen(). This is provided for consistency with relative fields, where the element returned by sage.rings.number_field.number_field_rel.NumberField_relative.gen() only generates the field over its base field (not necessarily over \(\QQ\)).

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 - 17)
sage: K.absolute_generator()
a

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(17), names=('a',)); (a,) = K._first_ngens(1)
>>> K.absolute_generator()
a

absolute_polynomial()[source]¶

Return absolute polynomial that defines this absolute field. This is the same as polynomial().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 + 1)
sage: K.absolute_polynomial ()
x^2 + 1

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.absolute_polynomial ()
x^2 + 1

absolute_vector_space(*args, **kwds)[source]¶

Return vector space over \(\QQ\) corresponding to this number field, along with maps from that space to this number field and in the other direction.

For an absolute extension this is identical to vector_space().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 - 5)
sage: K.absolute_vector_space()
(Vector space of dimension 3 over Rational Field,
 Isomorphism map:
  From: Vector space of dimension 3 over Rational Field
  To:   Number Field in a with defining polynomial x^3 - 5,
 Isomorphism map:
  From: Number Field in a with defining polynomial x^3 - 5
  To:   Vector space of dimension 3 over Rational Field)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> K.absolute_vector_space()
(Vector space of dimension 3 over Rational Field,
 Isomorphism map:
  From: Vector space of dimension 3 over Rational Field
  To:   Number Field in a with defining polynomial x^3 - 5,
 Isomorphism map:
  From: Number Field in a with defining polynomial x^3 - 5
  To:   Vector space of dimension 3 over Rational Field)

automorphisms()[source]¶

Compute all Galois automorphisms of self.

This uses PARI’s pari:nfgaloisconj and is much faster than root finding for many fields.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 + 10000)
sage: K.automorphisms()
[
Ring endomorphism of Number Field in a with defining polynomial x^2 + 10000
  Defn: a |--> a,
Ring endomorphism of Number Field in a with defining polynomial x^2 + 10000
  Defn: a |--> -a
]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(10000), names=('a',)); (a,) = K._first_ngens(1)
>>> K.automorphisms()
[
Ring endomorphism of Number Field in a with defining polynomial x^2 + 10000
  Defn: a |--> a,
Ring endomorphism of Number Field in a with defining polynomial x^2 + 10000
  Defn: a |--> -a
]

Here’s a larger example, that would take some time if we found roots instead of using PARI’s specialized machinery:

Sage

sage: K = NumberField(x^6 - x^4 - 2*x^2 + 1, 'a')
sage: len(K.automorphisms())
2

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(6) - x**Integer(4) - Integer(2)*x**Integer(2) + Integer(1), 'a')
>>> len(K.automorphisms())
2

\(L\) is the Galois closure of \(K\):

Sage

sage: L = NumberField(x^24 - 84*x^22 + 2814*x^20 - 15880*x^18 - 409563*x^16
....:                  - 8543892*x^14 + 25518202*x^12 + 32831026956*x^10
....:                  - 672691027218*x^8 - 4985379093428*x^6 + 320854419319140*x^4
....:                  + 817662865724712*x^2 + 513191437605441, 'a')
sage: len(L.automorphisms())
24

Python

>>> from sage.all import *
>>> L = NumberField(x**Integer(24) - Integer(84)*x**Integer(22) + Integer(2814)*x**Integer(20) - Integer(15880)*x**Integer(18) - Integer(409563)*x**Integer(16)
...                  - Integer(8543892)*x**Integer(14) + Integer(25518202)*x**Integer(12) + Integer(32831026956)*x**Integer(10)
...                  - Integer(672691027218)*x**Integer(8) - Integer(4985379093428)*x**Integer(6) + Integer(320854419319140)*x**Integer(4)
...                  + Integer(817662865724712)*x**Integer(2) + Integer(513191437605441), 'a')
>>> len(L.automorphisms())
24

Number fields defined by non-monic and non-integral polynomials are supported (Issue #252):

Sage

sage: R.<x> = QQ[]
sage: f = 7/9*x^3 + 7/3*x^2 - 56*x + 123
sage: K.<a> = NumberField(f)
sage: A = K.automorphisms(); A
[
Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
  Defn: a |--> a,
Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
  Defn: a |--> -7/15*a^2 - 18/5*a + 96/5,
Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
  Defn: a |--> 7/15*a^2 + 13/5*a - 111/5
]
sage: prod(x - sigma(a) for sigma in A) == f.monic()
True

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = Integer(7)/Integer(9)*x**Integer(3) + Integer(7)/Integer(3)*x**Integer(2) - Integer(56)*x + Integer(123)
>>> K = NumberField(f, names=('a',)); (a,) = K._first_ngens(1)
>>> A = K.automorphisms(); A
[
Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
  Defn: a |--> a,
Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
  Defn: a |--> -7/15*a^2 - 18/5*a + 96/5,
Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
  Defn: a |--> 7/15*a^2 + 13/5*a - 111/5
]
>>> prod(x - sigma(a) for sigma in A) == f.monic()
True

base_field()[source]¶

Return the base field of self, which is always QQ.

EXAMPLES:

Sage

sage: K = CyclotomicField(5)
sage: K.base_field()
Rational Field

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(5))
>>> K.base_field()
Rational Field

change_names(names)[source]¶

Return number field isomorphic to self but with the given generator name.

INPUT:

names – should be exactly one variable name

Also, K.structure() returns from_K and to_K, where from_K is an isomorphism from \(K\) to self and to_K is an isomorphism from self to \(K\).

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<z> = NumberField(x^2 + 3); K
Number Field in z with defining polynomial x^2 + 3
sage: L.<ww> = K.change_names()
sage: L
Number Field in ww with defining polynomial x^2 + 3
sage: L.structure()[0]
Isomorphism given by variable name change map:
  From: Number Field in ww with defining polynomial x^2 + 3
  To:   Number Field in z with defining polynomial x^2 + 3
sage: L.structure()[0](ww + 5/3)
z + 5/3

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(3), names=('z',)); (z,) = K._first_ngens(1); K
Number Field in z with defining polynomial x^2 + 3
>>> L = K.change_names(names=('ww',)); (ww,) = L._first_ngens(1)
>>> L
Number Field in ww with defining polynomial x^2 + 3
>>> L.structure()[Integer(0)]
Isomorphism given by variable name change map:
  From: Number Field in ww with defining polynomial x^2 + 3
  To:   Number Field in z with defining polynomial x^2 + 3
>>> L.structure()[Integer(0)](ww + Integer(5)/Integer(3))
z + 5/3

elements_of_bounded_height(**kwds)[source]¶

Return an iterator over the elements of self with relative multiplicative height at most bound.

This algorithm computes 2 lists: \(L\) containing elements \(x\) in \(K\) such that \(H_k(x) \leq B\), and a list \(L'\) containing elements \(x\) in \(K\) that, due to floating point issues, may be slightly larger than the bound. This can be controlled by lowering the tolerance.

In the current implementation, both lists \((L,L')\) are merged and returned in form of iterator.

ALGORITHM:

This is an implementation of the revised algorithm (Algorithm 4) in [DK2013]. Algorithm 5 is used for imaginary quadratic fields.

INPUT: keyword arguments:

bound – a real number
tolerance – (default: 0.01) a rational number in \((0,1]\)
precision – (default: 53) a positive integer

OUTPUT: an iterator of number field elements

EXAMPLES:

There are no elements in a number field with multiplicative height less than 1:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<g> = NumberField(x^5 - x + 19)
sage: list(K.elements_of_bounded_height(bound=0.9))
[]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(5) - x + Integer(19), names=('g',)); (g,) = K._first_ngens(1)
>>> list(K.elements_of_bounded_height(bound=RealNumber('0.9')))
[]

The only elements in a number field of height 1 are 0 and the roots of unity:

Sage

sage: K.<a> = NumberField(x^2 + x + 1)
sage: list(K.elements_of_bounded_height(bound=1))
[0, a + 1, a, -1, -a - 1, -a, 1]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> list(K.elements_of_bounded_height(bound=Integer(1)))
[0, a + 1, a, -1, -a - 1, -a, 1]

Sage

sage: K.<a> = CyclotomicField(20)
sage: len(list(K.elements_of_bounded_height(bound=1)))
21

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(20), names=('a',)); (a,) = K._first_ngens(1)
>>> len(list(K.elements_of_bounded_height(bound=Integer(1))))
21

The elements in the output iterator all have relative multiplicative height at most the input bound:

Sage

sage: K.<a> = NumberField(x^6 + 2)
sage: L = K.elements_of_bounded_height(bound=5)
sage: for t in L:
....:     exp(6*t.global_height())
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(6) + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.elements_of_bounded_height(bound=Integer(5))
>>> for t in L:
...     exp(Integer(6)*t.global_height())
1.00000000000000
1.00000000000000
1.00000000000000
2.00000000000000
2.00000000000000
2.00000000000000
2.00000000000000
4.00000000000000
4.00000000000000
4.00000000000000
4.00000000000000

Sage

sage: K.<a> = NumberField(x^2 - 71)
sage: L = K.elements_of_bounded_height(bound=20)
sage: all(exp(2*t.global_height()) <= 20 for t in L) # long time (5 s)
True

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) - Integer(71), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.elements_of_bounded_height(bound=Integer(20))
>>> all(exp(Integer(2)*t.global_height()) <= Integer(20) for t in L) # long time (5 s)
True

Sage

sage: K.<a> = NumberField(x^2 + 17)
sage: L = K.elements_of_bounded_height(bound=120)
sage: len(list(L))
9047

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(17), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.elements_of_bounded_height(bound=Integer(120))
>>> len(list(L))
9047

Sage

sage: K.<a> = NumberField(x^4 - 5)
sage: L = K.elements_of_bounded_height(bound=50)
sage: len(list(L)) # long time (2 s)
2163

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(4) - Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.elements_of_bounded_height(bound=Integer(50))
>>> len(list(L)) # long time (2 s)
2163

Sage

sage: K.<a> = CyclotomicField(13)
sage: L = K.elements_of_bounded_height(bound=2)
sage: len(list(L)) # long time (3 s)
27

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(13), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.elements_of_bounded_height(bound=Integer(2))
>>> len(list(L)) # long time (3 s)
27

Sage

sage: K.<a> = NumberField(x^6 + 2)
sage: L = K.elements_of_bounded_height(bound=60, precision=100)
sage: len(list(L)) # long time (5 s)
1899

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(6) + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.elements_of_bounded_height(bound=Integer(60), precision=Integer(100))
>>> len(list(L)) # long time (5 s)
1899

Sage

sage: K.<a> = NumberField(x^4 - x^3 - 3*x^2 + x + 1)
sage: L = K.elements_of_bounded_height(bound=10, tolerance=0.1)
sage: len(list(L))
99

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(4) - x**Integer(3) - Integer(3)*x**Integer(2) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.elements_of_bounded_height(bound=Integer(10), tolerance=RealNumber('0.1'))
>>> len(list(L))
99

AUTHORS:

John Doyle (2013)
David Krumm (2013)
Raman Raghukul (2018)

embeddings(K)[source]¶

Compute all field embeddings of this field into the field \(K\) (which need not even be a number field, e.g., it could be the complex numbers). This will return an identical result when given \(K\) as input again.

If possible, the most natural embedding of this field into \(K\) is put first in the list.

INPUT:

K – a field

EXAMPLES:

Sage

sage: # needs sage.groups
sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: L.<a1> = K.galois_closure(); L
Number Field in a1 with defining polynomial x^6 + 108
sage: K.embeddings(L)[0]
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Number Field in a1 with defining polynomial x^6 + 108
  Defn: a |--> 1/18*a1^4
sage: K.embeddings(L) is K.embeddings(L)
True

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.galois_closure(names=('a1',)); (a1,) = L._first_ngens(1); L
Number Field in a1 with defining polynomial x^6 + 108
>>> K.embeddings(L)[Integer(0)]
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Number Field in a1 with defining polynomial x^6 + 108
  Defn: a |--> 1/18*a1^4
>>> K.embeddings(L) is K.embeddings(L)
True

We embed a quadratic field into a cyclotomic field:

Sage

sage: L.<a> = QuadraticField(-7)
sage: K = CyclotomicField(7)
sage: L.embeddings(K)
[
Ring morphism:
  From: Number Field in a with defining polynomial x^2 + 7
        with a = 2.645751311064591?*I
  To:   Cyclotomic Field of order 7 and degree 6
  Defn: a |--> 2*zeta7^4 + 2*zeta7^2 + 2*zeta7 + 1,
Ring morphism:
  From: Number Field in a with defining polynomial x^2 + 7
        with a = 2.645751311064591?*I
  To:   Cyclotomic Field of order 7 and degree 6
  Defn: a |--> -2*zeta7^4 - 2*zeta7^2 - 2*zeta7 - 1
]

Python

>>> from sage.all import *
>>> L = QuadraticField(-Integer(7), names=('a',)); (a,) = L._first_ngens(1)
>>> K = CyclotomicField(Integer(7))
>>> L.embeddings(K)
[
Ring morphism:
  From: Number Field in a with defining polynomial x^2 + 7
        with a = 2.645751311064591?*I
  To:   Cyclotomic Field of order 7 and degree 6
  Defn: a |--> 2*zeta7^4 + 2*zeta7^2 + 2*zeta7 + 1,
Ring morphism:
  From: Number Field in a with defining polynomial x^2 + 7
        with a = 2.645751311064591?*I
  To:   Cyclotomic Field of order 7 and degree 6
  Defn: a |--> -2*zeta7^4 - 2*zeta7^2 - 2*zeta7 - 1
]

We embed a cubic field in the complex numbers:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: K.embeddings(CC)
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.62996052494743... - 1.09112363597172*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.62996052494743... + 1.09112363597172*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 1.25992104989487
]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.embeddings(CC)
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.62996052494743... - 1.09112363597172*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.62996052494743... + 1.09112363597172*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 - 2
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 1.25992104989487
]

Some more (possible and impossible) embeddings of cyclotomic fields:

Sage

sage: CyclotomicField(5).embeddings(QQbar)
[
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Algebraic Field
  Defn: zeta5 |--> 0.3090169943749474? + 0.9510565162951536?*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Algebraic Field
  Defn: zeta5 |--> -0.8090169943749474? + 0.5877852522924731?*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Algebraic Field
  Defn: zeta5 |--> -0.8090169943749474? - 0.5877852522924731?*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Algebraic Field
  Defn: zeta5 |--> 0.3090169943749474? - 0.9510565162951536?*I
]
sage: CyclotomicField(3).embeddings(CyclotomicField(7))
[ ]
sage: CyclotomicField(3).embeddings(CyclotomicField(6))
[
Ring morphism:
  From: Cyclotomic Field of order 3 and degree 2
  To:   Cyclotomic Field of order 6 and degree 2
  Defn: zeta3 |--> zeta6 - 1,
Ring morphism:
  From: Cyclotomic Field of order 3 and degree 2
  To:   Cyclotomic Field of order 6 and degree 2
  Defn: zeta3 |--> -zeta6
]

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(5)).embeddings(QQbar)
[
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Algebraic Field
  Defn: zeta5 |--> 0.3090169943749474? + 0.9510565162951536?*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Algebraic Field
  Defn: zeta5 |--> -0.8090169943749474? + 0.5877852522924731?*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Algebraic Field
  Defn: zeta5 |--> -0.8090169943749474? - 0.5877852522924731?*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Algebraic Field
  Defn: zeta5 |--> 0.3090169943749474? - 0.9510565162951536?*I
]
>>> CyclotomicField(Integer(3)).embeddings(CyclotomicField(Integer(7)))
[ ]
>>> CyclotomicField(Integer(3)).embeddings(CyclotomicField(Integer(6)))
[
Ring morphism:
  From: Cyclotomic Field of order 3 and degree 2
  To:   Cyclotomic Field of order 6 and degree 2
  Defn: zeta3 |--> zeta6 - 1,
Ring morphism:
  From: Cyclotomic Field of order 3 and degree 2
  To:   Cyclotomic Field of order 6 and degree 2
  Defn: zeta3 |--> -zeta6
]

Test that Issue #15053 is fixed:

Sage

sage: K = NumberField(x^3 - 2, 'a')
sage: K.embeddings(GF(3))
[]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) - Integer(2), 'a')
>>> K.embeddings(GF(Integer(3)))
[]

free_module(base=None, basis=None, map=True)[source]¶

Return a vector space \(V\) and isomorphisms self \(\to\) \(V\) and \(V\) \(\to\) self.

INPUT:

base – a subfield (default: None); the returned vector space is over this subfield \(R\), which defaults to the base field of this function field
basis – a basis for this field over the base
maps – boolean (default: True); whether to return \(R\)-linear maps to and from \(V\)

OUTPUT:

V – a vector space over the rational numbers
from_V – an isomorphism from \(V\) to self (if requested)
to_V – an isomorphism from self to \(V\) (if requested)

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^3 + 2)
sage: V, from_V, to_V  = k.free_module()
sage: from_V(V([1,2,3]))
3*a^2 + 2*a + 1
sage: to_V(1 + 2*a + 3*a^2)
(1, 2, 3)
sage: V
Vector space of dimension 3 over Rational Field
sage: to_V
Isomorphism map:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Vector space of dimension 3 over Rational Field
sage: from_V(to_V(2/3*a - 5/8))
2/3*a - 5/8
sage: to_V(from_V(V([0,-1/7,0])))
(0, -1/7, 0)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(3) + Integer(2), names=('a',)); (a,) = k._first_ngens(1)
>>> V, from_V, to_V  = k.free_module()
>>> from_V(V([Integer(1),Integer(2),Integer(3)]))
3*a^2 + 2*a + 1
>>> to_V(Integer(1) + Integer(2)*a + Integer(3)*a**Integer(2))
(1, 2, 3)
>>> V
Vector space of dimension 3 over Rational Field
>>> to_V
Isomorphism map:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Vector space of dimension 3 over Rational Field
>>> from_V(to_V(Integer(2)/Integer(3)*a - Integer(5)/Integer(8)))
2/3*a - 5/8
>>> to_V(from_V(V([Integer(0),-Integer(1)/Integer(7),Integer(0)])))
(0, -1/7, 0)

galois_closure(names=None, map=False)[source]¶

Return number field \(K\) that is the Galois closure of self, i.e., is generated by all roots of the defining polynomial of self, and possibly an embedding of self into \(K\).

INPUT:

names – variable name for Galois closure
map – boolean (default: False); also return an embedding of self into \(K\)

EXAMPLES:

Sage

sage: # needs sage.groups
sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^4 - 2)
sage: M = K.galois_closure('b'); M
Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
sage: L.<a2> = K.galois_closure(); L
Number Field in a2 with defining polynomial x^8 + 28*x^4 + 2500
sage: K.galois_group(names=("a3")).order()
8

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(4) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> M = K.galois_closure('b'); M
Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
>>> L = K.galois_closure(names=('a2',)); (a2,) = L._first_ngens(1); L
Number Field in a2 with defining polynomial x^8 + 28*x^4 + 2500
>>> K.galois_group(names=("a3")).order()
8

Sage

sage: # needs sage.groups
sage: phi = K.embeddings(L)[0]
sage: phi(K.0)
1/120*a2^5 + 19/60*a2
sage: phi(K.0).minpoly()
x^4 - 2

sage: # needs sage.groups
sage: L, phi = K.galois_closure('b', map=True)
sage: L
Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
sage: phi
Ring morphism:
  From: Number Field in a with defining polynomial x^4 - 2
  To:   Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
  Defn: a |--> 1/240*b^5 - 41/120*b

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> phi = K.embeddings(L)[Integer(0)]
>>> phi(K.gen(0))
1/120*a2^5 + 19/60*a2
>>> phi(K.gen(0)).minpoly()
x^4 - 2

>>> # needs sage.groups
>>> L, phi = K.galois_closure('b', map=True)
>>> L
Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
>>> phi
Ring morphism:
  From: Number Field in a with defining polynomial x^4 - 2
  To:   Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
  Defn: a |--> 1/240*b^5 - 41/120*b

A cyclotomic field is already Galois:

Sage

sage: # needs sage.groups
sage: K.<a> = NumberField(cyclotomic_polynomial(23))
sage: L.<z> = K.galois_closure()
sage: L
Number Field in z with defining polynomial
 x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12
  + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> K = NumberField(cyclotomic_polynomial(Integer(23)), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.galois_closure(names=('z',)); (z,) = L._first_ngens(1)
>>> L
Number Field in z with defining polynomial
 x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12
  + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

hilbert_conductor(a, b)[source]¶

This is the product of all (finite) primes where the Hilbert symbol is \(-1\). What is the same, this is the (reduced) discriminant of the quaternion algebra \((a,b)\) over a number field.

INPUT:

a, b – elements of the number field self

OUTPUT: squarefree ideal of the ring of integers of self

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: F.<a> = NumberField(x^2 - x - 1)
sage: F.hilbert_conductor(2*a, F(-1))
Fractional ideal (2)
sage: K.<b> = NumberField(x^3 - 4*x + 2)
sage: K.hilbert_conductor(K(2), K(-2))
Fractional ideal (1)
sage: K.hilbert_conductor(K(2*b), K(-2))
Fractional ideal (b^2 + b - 2)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> F = NumberField(x**Integer(2) - x - Integer(1), names=('a',)); (a,) = F._first_ngens(1)
>>> F.hilbert_conductor(Integer(2)*a, F(-Integer(1)))
Fractional ideal (2)
>>> K = NumberField(x**Integer(3) - Integer(4)*x + Integer(2), names=('b',)); (b,) = K._first_ngens(1)
>>> K.hilbert_conductor(K(Integer(2)), K(-Integer(2)))
Fractional ideal (1)
>>> K.hilbert_conductor(K(Integer(2)*b), K(-Integer(2)))
Fractional ideal (b^2 + b - 2)

AUTHOR:

Aly Deines

hilbert_symbol(a, b, P=None)[source]¶

Return the Hilbert symbol \((a,b)_P\) for a prime \(P\) of self and nonzero elements \(a\) and \(b\) of self.

If \(P\) is omitted, return the global Hilbert symbol \((a,b)\) instead.

INPUT:

a, b – elements of self
P – (default: None) if None, compute the global symbol. Otherwise, \(P\) should be either a prime ideal of self (which may also be given as a generator or set of generators) or a real or complex embedding.

OUTPUT: if \(a\) or \(b\) is zero, returns 0

If \(a\) and \(b\) are nonzero and \(P\) is specified, returns the Hilbert symbol \((a,b)_P\), which is \(1\) if the equation \(a x^2 + b y^2 = 1\) has a solution in the completion of self at \(P\), and is \(-1\) otherwise.

If \(a\) and \(b\) are nonzero and \(P\) is unspecified, returns \(1\) if the equation has a solution in self and \(-1\) otherwise.

EXAMPLES:

Some global examples:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 - 23)
sage: K.hilbert_symbol(0, a + 5)
0
sage: K.hilbert_symbol(a, 0)
0
sage: K.hilbert_symbol(-a, a + 1)
1
sage: K.hilbert_symbol(-a, a + 2)
-1
sage: K.hilbert_symbol(a, a + 5)
-1

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> K.hilbert_symbol(Integer(0), a + Integer(5))
0
>>> K.hilbert_symbol(a, Integer(0))
0
>>> K.hilbert_symbol(-a, a + Integer(1))
1
>>> K.hilbert_symbol(-a, a + Integer(2))
-1
>>> K.hilbert_symbol(a, a + Integer(5))
-1

That the latter two are unsolvable should be visible in local obstructions. For the first, this is a prime ideal above 19. For the second, the ramified prime above 23:

Sage

sage: K.hilbert_symbol(-a, a + 2, a + 2)
-1
sage: K.hilbert_symbol(a, a + 5, K.ideal(23).factor()[0][0])
-1

Python

>>> from sage.all import *
>>> K.hilbert_symbol(-a, a + Integer(2), a + Integer(2))
-1
>>> K.hilbert_symbol(a, a + Integer(5), K.ideal(Integer(23)).factor()[Integer(0)][Integer(0)])
-1

More local examples:

Sage

sage: K.hilbert_symbol(a, 0, K.fractional_ideal(5))
0
sage: K.hilbert_symbol(a, a + 5, K.fractional_ideal(5))
1
sage: K.hilbert_symbol(a + 1, 13, (a+6)*K)
-1
sage: [emb1, emb2] = K.embeddings(AA)
sage: K.hilbert_symbol(a, -1, emb1)
-1
sage: K.hilbert_symbol(a, -1, emb2)
1

Python

>>> from sage.all import *
>>> K.hilbert_symbol(a, Integer(0), K.fractional_ideal(Integer(5)))
0
>>> K.hilbert_symbol(a, a + Integer(5), K.fractional_ideal(Integer(5)))
1
>>> K.hilbert_symbol(a + Integer(1), Integer(13), (a+Integer(6))*K)
-1
>>> [emb1, emb2] = K.embeddings(AA)
>>> K.hilbert_symbol(a, -Integer(1), emb1)
-1
>>> K.hilbert_symbol(a, -Integer(1), emb2)
1

Ideals P can be given by generators:

Sage

sage: K.<a> = NumberField(x^5 - 23)
sage: pi = 2*a^4 + 3*a^3 + 4*a^2 + 15*a + 11
sage: K.hilbert_symbol(a, a + 5, pi)
1
sage: rho = 2*a^4 + 3*a^3 + 4*a^2 + 15*a + 11
sage: K.hilbert_symbol(a, a + 5, rho)
1

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(5) - Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> pi = Integer(2)*a**Integer(4) + Integer(3)*a**Integer(3) + Integer(4)*a**Integer(2) + Integer(15)*a + Integer(11)
>>> K.hilbert_symbol(a, a + Integer(5), pi)
1
>>> rho = Integer(2)*a**Integer(4) + Integer(3)*a**Integer(3) + Integer(4)*a**Integer(2) + Integer(15)*a + Integer(11)
>>> K.hilbert_symbol(a, a + Integer(5), rho)
1

This also works for non-principal ideals:

Sage

sage: K.<a> = QuadraticField(-5)
sage: P = K.ideal(3).factor()[0][0]
sage: P.gens_reduced()  # random, could be the other factor
(3, a + 1)
sage: K.hilbert_symbol(a, a + 3, P)
1
sage: K.hilbert_symbol(a, a + 3, [3, a+1])
1

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(3)).factor()[Integer(0)][Integer(0)]
>>> P.gens_reduced()  # random, could be the other factor
(3, a + 1)
>>> K.hilbert_symbol(a, a + Integer(3), P)
1
>>> K.hilbert_symbol(a, a + Integer(3), [Integer(3), a+Integer(1)])
1

Primes above 2:

Sage

sage: K.<a> = NumberField(x^5 - 23)
sage: p = [p[0] for p in (2*K).factor() if p[0].norm() == 16][0]
sage: K.hilbert_symbol(a, a + 5, p)
1
sage: K.hilbert_symbol(a, 2, p)
1
sage: K.hilbert_symbol(-1, a - 2, p)
-1

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(5) - Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> p = [p[Integer(0)] for p in (Integer(2)*K).factor() if p[Integer(0)].norm() == Integer(16)][Integer(0)]
>>> K.hilbert_symbol(a, a + Integer(5), p)
1
>>> K.hilbert_symbol(a, Integer(2), p)
1
>>> K.hilbert_symbol(-Integer(1), a - Integer(2), p)
-1

Various real fields are allowed:

Sage

sage: K.<a> = NumberField(x^3+x+1)
sage: K.hilbert_symbol(a/3, 1/2, K.embeddings(RDF)[0])
1
sage: K.hilbert_symbol(a/5, -1, K.embeddings(RR)[0])
-1
sage: [K.hilbert_symbol(a, -1, e) for e in K.embeddings(AA)]
[-1]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3)+x+Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.hilbert_symbol(a/Integer(3), Integer(1)/Integer(2), K.embeddings(RDF)[Integer(0)])
1
>>> K.hilbert_symbol(a/Integer(5), -Integer(1), K.embeddings(RR)[Integer(0)])
-1
>>> [K.hilbert_symbol(a, -Integer(1), e) for e in K.embeddings(AA)]
[-1]

Real embeddings are not allowed to be disguised as complex embeddings:

Sage

sage: K.<a> = QuadraticField(5)
sage: K.hilbert_symbol(-1, -1, K.embeddings(CC)[0])
Traceback (most recent call last):
...
ValueError: Possibly real place (=Ring morphism:
  From: Number Field in a with defining polynomial x^2 - 5
        with a = 2.236067977499790?
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -2.23606797749979)
given as complex embedding in hilbert_symbol. Is it real or complex?
sage: K.hilbert_symbol(-1, -1, K.embeddings(QQbar)[0])
Traceback (most recent call last):
...
ValueError: Possibly real place (=Ring morphism:
  From: Number Field in a with defining polynomial x^2 - 5
        with a = 2.236067977499790?
  To:   Algebraic Field
  Defn: a |--> -2.236067977499790?)
given as complex embedding in hilbert_symbol. Is it real or complex?
sage: K.<b> = QuadraticField(-5)
sage: K.hilbert_symbol(-1, -1, K.embeddings(CDF)[0])
1
sage: K.hilbert_symbol(-1, -1, K.embeddings(QQbar)[0])
1

Python

>>> from sage.all import *
>>> K = QuadraticField(Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> K.hilbert_symbol(-Integer(1), -Integer(1), K.embeddings(CC)[Integer(0)])
Traceback (most recent call last):
...
ValueError: Possibly real place (=Ring morphism:
  From: Number Field in a with defining polynomial x^2 - 5
        with a = 2.236067977499790?
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -2.23606797749979)
given as complex embedding in hilbert_symbol. Is it real or complex?
>>> K.hilbert_symbol(-Integer(1), -Integer(1), K.embeddings(QQbar)[Integer(0)])
Traceback (most recent call last):
...
ValueError: Possibly real place (=Ring morphism:
  From: Number Field in a with defining polynomial x^2 - 5
        with a = 2.236067977499790?
  To:   Algebraic Field
  Defn: a |--> -2.236067977499790?)
given as complex embedding in hilbert_symbol. Is it real or complex?
>>> K = QuadraticField(-Integer(5), names=('b',)); (b,) = K._first_ngens(1)
>>> K.hilbert_symbol(-Integer(1), -Integer(1), K.embeddings(CDF)[Integer(0)])
1
>>> K.hilbert_symbol(-Integer(1), -Integer(1), K.embeddings(QQbar)[Integer(0)])
1

\(a\) and \(b\) do not have to be integral or coprime:

Sage

sage: K.<i> = QuadraticField(-1)
sage: K.hilbert_symbol(1/2, 1/6, 3*K)
1
sage: p = 1 + i
sage: K.hilbert_symbol(p, p, p)
1
sage: K.hilbert_symbol(p, 3*p, p)
-1
sage: K.hilbert_symbol(3, p, p)
-1
sage: K.hilbert_symbol(1/3, 1/5, 1 + i)
1
sage: L = QuadraticField(5, 'a')
sage: L.hilbert_symbol(-3, -1/2, 2)
1

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.hilbert_symbol(Integer(1)/Integer(2), Integer(1)/Integer(6), Integer(3)*K)
1
>>> p = Integer(1) + i
>>> K.hilbert_symbol(p, p, p)
1
>>> K.hilbert_symbol(p, Integer(3)*p, p)
-1
>>> K.hilbert_symbol(Integer(3), p, p)
-1
>>> K.hilbert_symbol(Integer(1)/Integer(3), Integer(1)/Integer(5), Integer(1) + i)
1
>>> L = QuadraticField(Integer(5), 'a')
>>> L.hilbert_symbol(-Integer(3), -Integer(1)/Integer(2), Integer(2))
1

Various other examples:

Sage

sage: K.<a> = NumberField(x^3 + x + 1)
sage: K.hilbert_symbol(-6912, 24, -a^2 - a - 2)
1
sage: K.<a> = NumberField(x^5 - 23)
sage: P = K.ideal(-1105*a^4 + 1541*a^3 - 795*a^2 - 2993*a + 11853)
sage: Q = K.ideal(-7*a^4 + 13*a^3 - 13*a^2 - 2*a + 50)
sage: b = -a+5
sage: K.hilbert_symbol(a, b, P)
1
sage: K.hilbert_symbol(a, b, Q)
1
sage: K.<a> = NumberField(x^5 - 23)
sage: P = K.ideal(-1105*a^4 + 1541*a^3 - 795*a^2 - 2993*a + 11853)
sage: K.hilbert_symbol(a, a + 5, P)
1
sage: K.hilbert_symbol(a, 2, P)
1
sage: K.hilbert_symbol(a + 5, 2, P)
-1
sage: K.<a> = NumberField(x^3 - 4*x + 2)
sage: K.hilbert_symbol(2, -2, K.primes_above(2)[0])
1

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.hilbert_symbol(-Integer(6912), Integer(24), -a**Integer(2) - a - Integer(2))
1
>>> K = NumberField(x**Integer(5) - Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(-Integer(1105)*a**Integer(4) + Integer(1541)*a**Integer(3) - Integer(795)*a**Integer(2) - Integer(2993)*a + Integer(11853))
>>> Q = K.ideal(-Integer(7)*a**Integer(4) + Integer(13)*a**Integer(3) - Integer(13)*a**Integer(2) - Integer(2)*a + Integer(50))
>>> b = -a+Integer(5)
>>> K.hilbert_symbol(a, b, P)
1
>>> K.hilbert_symbol(a, b, Q)
1
>>> K = NumberField(x**Integer(5) - Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(-Integer(1105)*a**Integer(4) + Integer(1541)*a**Integer(3) - Integer(795)*a**Integer(2) - Integer(2993)*a + Integer(11853))
>>> K.hilbert_symbol(a, a + Integer(5), P)
1
>>> K.hilbert_symbol(a, Integer(2), P)
1
>>> K.hilbert_symbol(a + Integer(5), Integer(2), P)
-1
>>> K = NumberField(x**Integer(3) - Integer(4)*x + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.hilbert_symbol(Integer(2), -Integer(2), K.primes_above(Integer(2))[Integer(0)])
1

Check that the bug reported at Issue #16043 has been fixed:

Sage

sage: K.<a> = NumberField(x^2 + 5)
sage: p = K.primes_above(2)[0]; p
Fractional ideal (2, a + 1)
sage: K.hilbert_symbol(2*a, -1, p)
1
sage: K.hilbert_symbol(2*a, 2, p)
-1
sage: K.hilbert_symbol(2*a, -2, p)
-1

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> p = K.primes_above(Integer(2))[Integer(0)]; p
Fractional ideal (2, a + 1)
>>> K.hilbert_symbol(Integer(2)*a, -Integer(1), p)
1
>>> K.hilbert_symbol(Integer(2)*a, Integer(2), p)
-1
>>> K.hilbert_symbol(Integer(2)*a, -Integer(2), p)
-1

AUTHOR:

Aly Deines (2010-08-19): part of the doctests
Marco Streng (2010-12-06)

hilbert_symbol_negative_at_S(S, b, check=True)[source]¶

Return \(a\) such that the Hilbert conductor of \(a\) and \(b\) is \(S\).

INPUT:

S – list of places (or prime ideals) of even cardinality
b – a nonzero rational number which is a non-square locally at every place in \(S\)
check – boolean (default: True); perform additional checks on the input and confirm the output

OUTPUT:

an element \(a\) that has negative Hilbert symbol \((a,b)_p\) for every (finite and infinite) place \(p\) in \(S\).

ALGORITHM:

The implementation is following algorithm 3.4.1 in [Kir2016]. We note that class and unit groups are computed using the generalized Riemann hypothesis. If it is false, this may result in an infinite loop. Nevertheless, if the algorithm terminates the output is correct.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 + 20072)
sage: S = [K.primes_above(3)[0], K.primes_above(23)[0]]
sage: b = K.hilbert_symbol_negative_at_S(S, a + 1)
sage: [K.hilbert_symbol(b, a + 1, p) for p in S]
[-1, -1]
sage: K.<d> = CyclotomicField(11)
sage: S = [K.primes_above(2)[0], K.primes_above(11)[0]]
sage: b = d + 5
sage: a = K.hilbert_symbol_negative_at_S(S, b)
sage: [K.hilbert_symbol(a,b,p) for p in S]
[-1, -1]
sage: k.<c> = K.maximal_totally_real_subfield()[0]
sage: S = [k.primes_above(3)[0], k.primes_above(5)[0]]
sage: S += k.real_places()[:2]
sage: b = 5 + c + c^9
sage: a = k.hilbert_symbol_negative_at_S(S, b)
sage: [k.hilbert_symbol(a, b, p) for p in S]
[-1, -1, -1, -1]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(20072), names=('a',)); (a,) = K._first_ngens(1)
>>> S = [K.primes_above(Integer(3))[Integer(0)], K.primes_above(Integer(23))[Integer(0)]]
>>> b = K.hilbert_symbol_negative_at_S(S, a + Integer(1))
>>> [K.hilbert_symbol(b, a + Integer(1), p) for p in S]
[-1, -1]
>>> K = CyclotomicField(Integer(11), names=('d',)); (d,) = K._first_ngens(1)
>>> S = [K.primes_above(Integer(2))[Integer(0)], K.primes_above(Integer(11))[Integer(0)]]
>>> b = d + Integer(5)
>>> a = K.hilbert_symbol_negative_at_S(S, b)
>>> [K.hilbert_symbol(a,b,p) for p in S]
[-1, -1]
>>> k = K.maximal_totally_real_subfield()[Integer(0)]; (c,) = k._first_ngens(1)
>>> S = [k.primes_above(Integer(3))[Integer(0)], k.primes_above(Integer(5))[Integer(0)]]
>>> S += k.real_places()[:Integer(2)]
>>> b = Integer(5) + c + c**Integer(9)
>>> a = k.hilbert_symbol_negative_at_S(S, b)
>>> [k.hilbert_symbol(a, b, p) for p in S]
[-1, -1, -1, -1]

Note that the closely related Hilbert conductor takes only the finite places into account:

Sage

sage: k.hilbert_conductor(a, b)
Fractional ideal (15)

Python

>>> from sage.all import *
>>> k.hilbert_conductor(a, b)
Fractional ideal (15)

AUTHORS:

Simon Brandhorst, Anna Haensch (01-05-2018)

is_absolute()[source]¶

Return True since self is an absolute field.

EXAMPLES:

Sage

sage: K = CyclotomicField(5)
sage: K.is_absolute()
True

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(5))
>>> K.is_absolute()
True

logarithmic_embedding(prec=53)[source]¶

Return the morphism of self under the logarithmic embedding in the category Set.

The logarithmic embedding is defined as a map from the number field self to \(\RR^n\).

It is defined under Definition 4.9.6 in [Coh1993].

INPUT:

prec – desired floating point precision

OUTPUT: the morphism of self under the logarithmic embedding in the category Set

EXAMPLES:

Sage

sage: CF.<a> = CyclotomicField(5)
sage: f = CF.logarithmic_embedding()
sage: f(0)
(-1, -1)
sage: f(7)
(3.89182029811063, 3.89182029811063)

Python

>>> from sage.all import *
>>> CF = CyclotomicField(Integer(5), names=('a',)); (a,) = CF._first_ngens(1)
>>> f = CF.logarithmic_embedding()
>>> f(Integer(0))
(-1, -1)
>>> f(Integer(7))
(3.89182029811063, 3.89182029811063)

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 + 5)
sage: f = K.logarithmic_embedding()
sage: f(0)
(-1, -1)
sage: f(7)
(1.94591014905531, 3.89182029811063)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) + Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> f = K.logarithmic_embedding()
>>> f(Integer(0))
(-1, -1)
>>> f(Integer(7))
(1.94591014905531, 3.89182029811063)

Sage

sage: F.<a> = NumberField(x^4 - 8*x^2 + 3)
sage: f = F.logarithmic_embedding()
sage: f(0)
(-1, -1, -1, -1)
sage: f(7)
(1.94591014905531, 1.94591014905531, 1.94591014905531, 1.94591014905531)

Python

>>> from sage.all import *
>>> F = NumberField(x**Integer(4) - Integer(8)*x**Integer(2) + Integer(3), names=('a',)); (a,) = F._first_ngens(1)
>>> f = F.logarithmic_embedding()
>>> f(Integer(0))
(-1, -1, -1, -1)
>>> f(Integer(7))
(1.94591014905531, 1.94591014905531, 1.94591014905531, 1.94591014905531)

minkowski_embedding(B=None, prec=None)[source]¶

Return an \(n \times n\) matrix over RDF whose columns are the images of the basis \(\{1, \alpha, \dots, \alpha^{n-1}\}\) of self over \(\QQ\) (as vector spaces), where here \(\alpha\) is the generator of self over \(\QQ\), i.e. self.gen(0). If \(B\) is not None, return the images of the vectors in \(B\) as the columns instead. If prec is not None, use RealField(prec) instead of RDF.

This embedding is the so-called “Minkowski embedding” of a number field in \(\RR^n\): given the \(n\) embeddings \(\sigma_1, \dots, \sigma_n\) of self in \(\CC\), write \(\sigma_1, \dots, \sigma_r\) for the real embeddings, and \(\sigma_{r+1}, \dots, \sigma_{r+s}\) for choices of one of each pair of complex conjugate embeddings (in our case, we simply choose the one where the image of \(\alpha\) has positive real part). Here \((r,s)\) is the signature of self. Then the Minkowski embedding is given by

\[x \mapsto ( \sigma_1(x), \dots, \sigma_r(x), \sqrt{2}\Re(\sigma_{r+1}(x)), \sqrt{2}\Im(\sigma_{r+1}(x)), \dots, \sqrt{2}\Re(\sigma_{r+s}(x)), \sqrt{2}\Im(\sigma_{r+s}(x)))\]

Equivalently, this is an embedding of self in \(\RR^n\) so that the usual norm on \(\RR^n\) coincides with \(|x| = \sum_i |\sigma_i(x)|^2\) on self.

Todo

This could be much improved by implementing homomorphisms over VectorSpaces.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: F.<alpha> = NumberField(x^3 + 2)
sage: F.minkowski_embedding()
[ 1.00000000000000 -1.25992104989487  1.58740105196820]
[ 1.41421356237... 0.8908987181... -1.12246204830...]
[0.000000000000000  1.54308184421...  1.94416129723...]
sage: F.minkowski_embedding([1, alpha+2, alpha^2-alpha])
[ 1.00000000000000 0.740078950105127  2.84732210186307]
[ 1.41421356237...  3.7193258428... -2.01336076644...]
[0.000000000000000  1.54308184421... 0.40107945302...]
sage: F.minkowski_embedding() * (alpha + 2).vector().column()
[0.740078950105127]
[ 3.7193258428...]
[ 1.54308184421...]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> F = NumberField(x**Integer(3) + Integer(2), names=('alpha',)); (alpha,) = F._first_ngens(1)
>>> F.minkowski_embedding()
[ 1.00000000000000 -1.25992104989487  1.58740105196820]
[ 1.41421356237... 0.8908987181... -1.12246204830...]
[0.000000000000000  1.54308184421...  1.94416129723...]
>>> F.minkowski_embedding([Integer(1), alpha+Integer(2), alpha**Integer(2)-alpha])
[ 1.00000000000000 0.740078950105127  2.84732210186307]
[ 1.41421356237...  3.7193258428... -2.01336076644...]
[0.000000000000000  1.54308184421... 0.40107945302...]
>>> F.minkowski_embedding() * (alpha + Integer(2)).vector().column()
[0.740078950105127]
[ 3.7193258428...]
[ 1.54308184421...]

optimized_representation(name=None, both_maps=True)[source]¶

Return a field isomorphic to self with a better defining polynomial if possible, along with field isomorphisms from the new field to self and from self to the new field.

EXAMPLES: We construct a compositum of 3 quadratic fields, then find an optimized representation and transform elements back and forth.

Sage

sage: x = polygen(QQ, 'x')
sage: K = NumberField([x^2 + p for p in [5, 3, 2]],'a').absolute_field('b'); K
Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576
sage: L, from_L, to_L = K.optimized_representation()
sage: L    # your answer may different, since algorithm is random
Number Field in b1 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81
sage: to_L(K.0)   # random
4/189*b1^7 + 1/63*b1^6 + 1/27*b1^5 - 2/9*b1^4 - 5/27*b1^3 - 8/9*b1^2 + 3/7*b1 - 3/7
sage: from_L(L.0)   # random
1/1152*b^7 - 1/192*b^6 + 23/576*b^5 - 17/96*b^4 + 37/72*b^3 - 5/6*b^2 + 55/24*b - 3/4

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField([x**Integer(2) + p for p in [Integer(5), Integer(3), Integer(2)]],'a').absolute_field('b'); K
Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576
>>> L, from_L, to_L = K.optimized_representation()
>>> L    # your answer may different, since algorithm is random
Number Field in b1 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81
>>> to_L(K.gen(0))   # random
4/189*b1^7 + 1/63*b1^6 + 1/27*b1^5 - 2/9*b1^4 - 5/27*b1^3 - 8/9*b1^2 + 3/7*b1 - 3/7
>>> from_L(L.gen(0))   # random
1/1152*b^7 - 1/192*b^6 + 23/576*b^5 - 17/96*b^4 + 37/72*b^3 - 5/6*b^2 + 55/24*b - 3/4

The transformation maps are mutually inverse isomorphisms.

Sage

sage: from_L(to_L(K.0)) == K.0
True
sage: to_L(from_L(L.0)) == L.0
True

Python

>>> from sage.all import *
>>> from_L(to_L(K.gen(0))) == K.gen(0)
True
>>> to_L(from_L(L.gen(0))) == L.gen(0)
True

Number fields defined by non-monic and non-integral polynomials are supported (Issue #252):

Sage

sage: K.<a> = NumberField(7/9*x^3 + 7/3*x^2 - 56*x + 123)
sage: K.optimized_representation()  # representation varies, not tested
(Number Field in a1 with defining polynomial x^3 - 7*x - 7,
 Ring morphism:
   From: Number Field in a1 with defining polynomial x^3 - 7*x - 7
   To:   Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
   Defn: a1 |--> 7/225*a^2 - 7/75*a - 42/25,
 Ring morphism:
   From: Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
   To:   Number Field in a1 with defining polynomial x^3 - 7*x - 7
   Defn: a |--> -15/7*a1^2 + 9)

Python

>>> from sage.all import *
>>> K = NumberField(Integer(7)/Integer(9)*x**Integer(3) + Integer(7)/Integer(3)*x**Integer(2) - Integer(56)*x + Integer(123), names=('a',)); (a,) = K._first_ngens(1)
>>> K.optimized_representation()  # representation varies, not tested
(Number Field in a1 with defining polynomial x^3 - 7*x - 7,
 Ring morphism:
   From: Number Field in a1 with defining polynomial x^3 - 7*x - 7
   To:   Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
   Defn: a1 |--> 7/225*a^2 - 7/75*a - 42/25,
 Ring morphism:
   From: Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x + 123
   To:   Number Field in a1 with defining polynomial x^3 - 7*x - 7
   Defn: a |--> -15/7*a1^2 + 9)

optimized_subfields(degree=0, name=None, both_maps=True)[source]¶

Return optimized representations of many (but not necessarily all!) subfields of self of the given degree, or of all possible degrees if degree is 0.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K = NumberField([x^2 + p for p in [5, 3, 2]],'a').absolute_field('b'); K
Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576
sage: L = K.optimized_subfields(name='b')
sage: L[0][0]
Number Field in b0 with defining polynomial x
sage: L[1][0]
Number Field in b1 with defining polynomial x^2 - 3*x + 3
sage: [z[0] for z in L]          # random -- since algorithm is random
[Number Field in b0 with defining polynomial x - 1,
 Number Field in b1 with defining polynomial x^2 - x + 1,
 Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25,
 Number Field in b3 with defining polynomial x^4 - 2*x^2 + 4,
 Number Field in b4 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField([x**Integer(2) + p for p in [Integer(5), Integer(3), Integer(2)]],'a').absolute_field('b'); K
Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576
>>> L = K.optimized_subfields(name='b')
>>> L[Integer(0)][Integer(0)]
Number Field in b0 with defining polynomial x
>>> L[Integer(1)][Integer(0)]
Number Field in b1 with defining polynomial x^2 - 3*x + 3
>>> [z[Integer(0)] for z in L]          # random -- since algorithm is random
[Number Field in b0 with defining polynomial x - 1,
 Number Field in b1 with defining polynomial x^2 - x + 1,
 Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25,
 Number Field in b3 with defining polynomial x^4 - 2*x^2 + 4,
 Number Field in b4 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81]

We examine one of the optimized subfields in more detail:

Sage

sage: M, from_M, to_M = L[2]
sage: M                             # random
Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25
sage: from_M     # may be slightly random
Ring morphism:
  From: Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25
  To:   Number Field in a1 with defining polynomial
        x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576
  Defn: b2 |--> -5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4
                - 95/72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1

Python

>>> from sage.all import *
>>> M, from_M, to_M = L[Integer(2)]
>>> M                             # random
Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25
>>> from_M     # may be slightly random
Ring morphism:
  From: Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25
  To:   Number Field in a1 with defining polynomial
        x^8 + 40*x^6 + 352*x^4 + 960*x^2 + 576
  Defn: b2 |--> -5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4
                - 95/72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1

The to_M map is None, since there is no map from \(K\) to \(M\):

Sage

sage: to_M

Python

>>> from sage.all import *
>>> to_M

We apply the from_M map to the generator of M, which gives a rather large element of \(K\):

Sage

sage: from_M(M.0)          # random
-5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4
 - 95/72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1

Python

>>> from sage.all import *
>>> from_M(M.gen(0))          # random
-5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4
 - 95/72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1

Nevertheless, that large-ish element lies in a degree 4 subfield:

Sage

sage: from_M(M.0).minpoly()   # random
x^4 - 5*x^2 + 25

Python

>>> from sage.all import *
>>> from_M(M.gen(0)).minpoly()   # random
x^4 - 5*x^2 + 25

order(*args, **kwds)[source]¶

Return the order with given ring generators in the maximal order of this number field.

INPUT:

gens – list of elements in this number field; if no generators are given, just returns the cardinality of this number field (\(\infty\)) for consistency.
check_is_integral – boolean (default: True); whether to check that each generator is integral
check_rank – boolean (default: True); whether to check that the ring generated by gens is of full rank
allow_subfield – boolean (default: False); if True and the generators do not generate an order, i.e., they generate a subring of smaller rank, instead of raising an error, return an order in a smaller number field

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<i> = NumberField(x^2 + 1)
sage: k.order(2*i)
Order of conductor 2 generated by 2*i in Number Field in i with defining polynomial x^2 + 1
sage: k.order(10*i)
Order of conductor 10 generated by 10*i in Number Field in i with defining polynomial x^2 + 1
sage: k.order(3)
Traceback (most recent call last):
...
ValueError: the rank of the span of gens is wrong
sage: k.order(i/2)
Traceback (most recent call last):
...
ValueError: each generator must be integral

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = k._first_ngens(1)
>>> k.order(Integer(2)*i)
Order of conductor 2 generated by 2*i in Number Field in i with defining polynomial x^2 + 1
>>> k.order(Integer(10)*i)
Order of conductor 10 generated by 10*i in Number Field in i with defining polynomial x^2 + 1
>>> k.order(Integer(3))
Traceback (most recent call last):
...
ValueError: the rank of the span of gens is wrong
>>> k.order(i/Integer(2))
Traceback (most recent call last):
...
ValueError: each generator must be integral

Alternatively, an order can be constructed by adjoining elements to \(\ZZ\):

Sage

sage: K.<a> = NumberField(x^3 - 2)
sage: ZZ[a]
Order generated by a0 in Number Field in a0 with defining polynomial x^3 - 2 with a0 = a

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> ZZ[a]
Order generated by a0 in Number Field in a0 with defining polynomial x^3 - 2 with a0 = a

places(all_complex=False, prec=None)[source]¶

Return the collection of all infinite places of self.

By default, this returns the set of real places as homomorphisms into RIF first, followed by a choice of one of each pair of complex conjugate homomorphisms into CIF.

On the other hand, if prec is not None, we simply return places into RealField(prec) and ComplexField(prec) (or RDF, CDF if prec=53). One can also use prec=infinity, which returns embeddings into the field \(\overline{\QQ}\) of algebraic numbers (or its subfield \(\mathbb{A}\) of algebraic reals); this permits exact computation, but can be extremely slow.

There is an optional flag all_complex, which defaults to False. If all_complex is True, then the real embeddings are returned as embeddings into CIF instead of RIF.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: F.<alpha> = NumberField(x^3 - 100*x + 1); F.places()
[Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> -10.00499625499181184573367219280,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> 0.01000001000003000012000055000273,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> 9.994996244991781845613530439509]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> F = NumberField(x**Integer(3) - Integer(100)*x + Integer(1), names=('alpha',)); (alpha,) = F._first_ngens(1); F.places()
[Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> -10.00499625499181184573367219280,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> 0.01000001000003000012000055000273,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> 9.994996244991781845613530439509]

Sage

sage: F.<alpha> = NumberField(x^3 + 7); F.places()
[Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> -1.912931182772389101199116839549,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Complex Field with 53 bits of precision
   Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]

Python

>>> from sage.all import *
>>> F = NumberField(x**Integer(3) + Integer(7), names=('alpha',)); (alpha,) = F._first_ngens(1); F.places()
[Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> -1.912931182772389101199116839549,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Complex Field with 53 bits of precision
   Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]

Sage

sage: F.<alpha> = NumberField(x^3 + 7) ; F.places(all_complex=True)
[Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Complex Field with 53 bits of precision
   Defn: alpha |--> -1.91293118277239,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Complex Field with 53 bits of precision
   Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]
sage: F.places(prec=10)
[Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Real Field with 10 bits of precision
   Defn: alpha |--> -1.9,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Complex Field with 10 bits of precision
   Defn: alpha |--> 0.96 + 1.7*I]

Python

>>> from sage.all import *
>>> F = NumberField(x**Integer(3) + Integer(7), names=('alpha',)); (alpha,) = F._first_ngens(1); F.places(all_complex=True)
[Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Complex Field with 53 bits of precision
   Defn: alpha |--> -1.91293118277239,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Complex Field with 53 bits of precision
   Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]
>>> F.places(prec=Integer(10))
[Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Real Field with 10 bits of precision
   Defn: alpha |--> -1.9,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^3 + 7
   To:   Complex Field with 10 bits of precision
   Defn: alpha |--> 0.96 + 1.7*I]

real_places(prec=None)[source]¶

Return all real places of self as homomorphisms into RIF.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: F.<alpha> = NumberField(x^4 - 7) ; F.real_places()
[Ring morphism:
   From: Number Field in alpha with defining polynomial x^4 - 7
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> -1.626576561697785743211232345494,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^4 - 7
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> 1.626576561697785743211232345494]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> F = NumberField(x**Integer(4) - Integer(7), names=('alpha',)); (alpha,) = F._first_ngens(1); F.real_places()
[Ring morphism:
   From: Number Field in alpha with defining polynomial x^4 - 7
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> -1.626576561697785743211232345494,
 Ring morphism:
   From: Number Field in alpha with defining polynomial x^4 - 7
   To:   Real Field with 106 bits of precision
   Defn: alpha |--> 1.626576561697785743211232345494]

relative_degree()[source]¶

A synonym for degree().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_degree()
2

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.relative_degree()
2

relative_different()[source]¶

A synonym for different().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_different()
Fractional ideal (2)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.relative_different()
Fractional ideal (2)

relative_discriminant()[source]¶

A synonym for discriminant().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_discriminant()
-4

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.relative_discriminant()
-4

relative_polynomial()[source]¶

A synonym for polynomial().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_polynomial()
x^2 + 1

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.relative_polynomial()
x^2 + 1

relative_vector_space(*args, **kwds)[source]¶

A synonym for vector_space().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_vector_space()
(Vector space of dimension 2 over Rational Field,
 Isomorphism map:
  From: Vector space of dimension 2 over Rational Field
  To:   Number Field in i with defining polynomial x^2 + 1,
 Isomorphism map:
  From: Number Field in i with defining polynomial x^2 + 1
  To:   Vector space of dimension 2 over Rational Field)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.relative_vector_space()
(Vector space of dimension 2 over Rational Field,
 Isomorphism map:
  From: Vector space of dimension 2 over Rational Field
  To:   Number Field in i with defining polynomial x^2 + 1,
 Isomorphism map:
  From: Number Field in i with defining polynomial x^2 + 1
  To:   Vector space of dimension 2 over Rational Field)

relativize(alpha, names, structure=None)[source]¶

Given an element in self or an embedding of a subfield into self, return a relative number field \(K\) isomorphic to self that is relative over the absolute field \(\QQ(\alpha)\) or the domain of \(\alpha\), along with isomorphisms from \(K\) to self and from self to \(K\).

INPUT:

alpha – an element of self or an embedding of a subfield into self
names – 2-tuple of names of generator for output field \(K\) and the subfield \(\QQ(\alpha)\)
structure – an instance of structure.NumberFieldStructure or None (default: None), if None, then the resulting field’s structure() will return isomorphisms from and to this field. Otherwise, the field will be equipped with structure.

OUTPUT: \(K\) – relative number field

Also, K.structure() returns from_K and to_K, where from_K is an isomorphism from \(K\) to self and to_K is an isomorphism from self to \(K\).

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^10 - 2)
sage: L.<c,d> = K.relativize(a^4 + a^2 + 2); L
Number Field in c with defining polynomial
 x^2 - 1/5*d^4 + 8/5*d^3 - 23/5*d^2 + 7*d - 18/5 over its base field
sage: c.absolute_minpoly()
x^10 - 2
sage: d.absolute_minpoly()
x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58
sage: (a^4 + a^2 + 2).minpoly()
x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58
sage: from_L, to_L = L.structure()
sage: to_L(a)
c
sage: to_L(a^4 + a^2 + 2)
d
sage: from_L(to_L(a^4 + a^2 + 2))
a^4 + a^2 + 2

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(10) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.relativize(a**Integer(4) + a**Integer(2) + Integer(2), names=('c', 'd',)); (c, d,) = L._first_ngens(2); L
Number Field in c with defining polynomial
 x^2 - 1/5*d^4 + 8/5*d^3 - 23/5*d^2 + 7*d - 18/5 over its base field
>>> c.absolute_minpoly()
x^10 - 2
>>> d.absolute_minpoly()
x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58
>>> (a**Integer(4) + a**Integer(2) + Integer(2)).minpoly()
x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58
>>> from_L, to_L = L.structure()
>>> to_L(a)
c
>>> to_L(a**Integer(4) + a**Integer(2) + Integer(2))
d
>>> from_L(to_L(a**Integer(4) + a**Integer(2) + Integer(2)))
a^4 + a^2 + 2

The following demonstrates distinct embeddings of a subfield into a larger field:

Sage

sage: K.<a> = NumberField(x^4 + 2*x^2 + 2)
sage: K0 = K.subfields(2)[0][0]; K0
Number Field in a0 with defining polynomial x^2 - 2*x + 2
sage: rho, tau = K0.embeddings(K)
sage: L0 = K.relativize(rho(K0.gen()), 'b'); L0
Number Field in b0 with defining polynomial x^2 - b1 + 2 over its base field
sage: L1 = K.relativize(rho, 'b'); L1
Number Field in b with defining polynomial x^2 - a0 + 2 over its base field
sage: L2 = K.relativize(tau, 'b'); L2
Number Field in b with defining polynomial x^2 + a0 over its base field
sage: L0.base_field() is K0
False
sage: L1.base_field() is K0
True
sage: L2.base_field() is K0
True

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(4) + Integer(2)*x**Integer(2) + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K0 = K.subfields(Integer(2))[Integer(0)][Integer(0)]; K0
Number Field in a0 with defining polynomial x^2 - 2*x + 2
>>> rho, tau = K0.embeddings(K)
>>> L0 = K.relativize(rho(K0.gen()), 'b'); L0
Number Field in b0 with defining polynomial x^2 - b1 + 2 over its base field
>>> L1 = K.relativize(rho, 'b'); L1
Number Field in b with defining polynomial x^2 - a0 + 2 over its base field
>>> L2 = K.relativize(tau, 'b'); L2
Number Field in b with defining polynomial x^2 + a0 over its base field
>>> L0.base_field() is K0
False
>>> L1.base_field() is K0
True
>>> L2.base_field() is K0
True

Here we see that with the different embeddings, the relative norms are different:

Sage

sage: a0 = K0.gen()
sage: L1_into_K, K_into_L1 = L1.structure()
sage: L2_into_K, K_into_L2 = L2.structure()
sage: len(K.factor(41))
4
sage: w1 = -a^2 + a + 1; P = K.ideal([w1])
sage: Pp = L1.ideal(K_into_L1(w1)).ideal_below(); Pp == K0.ideal([4*a0 + 1])
True
sage: Pp == w1.norm(rho)
True

sage: w2 = a^2 + a - 1; Q = K.ideal([w2])
sage: Qq = L2.ideal(K_into_L2(w2)).ideal_below(); Qq == K0.ideal([-4*a0 + 9])
True
sage: Qq == w2.norm(tau)
True

sage: Pp == Qq
False

Python

>>> from sage.all import *
>>> a0 = K0.gen()
>>> L1_into_K, K_into_L1 = L1.structure()
>>> L2_into_K, K_into_L2 = L2.structure()
>>> len(K.factor(Integer(41)))
4
>>> w1 = -a**Integer(2) + a + Integer(1); P = K.ideal([w1])
>>> Pp = L1.ideal(K_into_L1(w1)).ideal_below(); Pp == K0.ideal([Integer(4)*a0 + Integer(1)])
True
>>> Pp == w1.norm(rho)
True

>>> w2 = a**Integer(2) + a - Integer(1); Q = K.ideal([w2])
>>> Qq = L2.ideal(K_into_L2(w2)).ideal_below(); Qq == K0.ideal([-Integer(4)*a0 + Integer(9)])
True
>>> Qq == w2.norm(tau)
True

>>> Pp == Qq
False

subfields(degree=0, name=None)[source]¶

Return all subfields of self of the given degree, or of all possible degrees if degree is 0. The subfields are returned as absolute fields together with an embedding into self. For the case of the field itself, the reverse isomorphism is also provided.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField([x^3 - 2, x^2 + x + 1])
sage: K = K.absolute_field('b')
sage: S = K.subfields()
sage: len(S)
6
sage: [k[0].polynomial() for k in S]
[x - 3,
 x^2 - 3*x + 9,
 x^3 - 3*x^2 + 3*x + 1,
 x^3 - 3*x^2 + 3*x + 1,
 x^3 - 3*x^2 + 3*x - 17,
 x^6 - 3*x^5 + 6*x^4 - 11*x^3 + 12*x^2 + 3*x + 1]
sage: R.<t> = QQ[]
sage: L = NumberField(t^3 - 3*t + 1, 'c')
sage: [k[1] for k in L.subfields()]
[Ring morphism:
  From: Number Field in c0 with defining polynomial t
  To:   Number Field in c with defining polynomial t^3 - 3*t + 1
  Defn: 0 |--> 0,
 Ring morphism:
  From: Number Field in c1 with defining polynomial t^3 - 3*t + 1
  To:   Number Field in c with defining polynomial t^3 - 3*t + 1
  Defn: c1 |--> c]
sage: len(L.subfields(2))
0
sage: len(L.subfields(1))
1

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField([x**Integer(3) - Integer(2), x**Integer(2) + x + Integer(1)], names=('a',)); (a,) = K._first_ngens(1)
>>> K = K.absolute_field('b')
>>> S = K.subfields()
>>> len(S)
6
>>> [k[Integer(0)].polynomial() for k in S]
[x - 3,
 x^2 - 3*x + 9,
 x^3 - 3*x^2 + 3*x + 1,
 x^3 - 3*x^2 + 3*x + 1,
 x^3 - 3*x^2 + 3*x - 17,
 x^6 - 3*x^5 + 6*x^4 - 11*x^3 + 12*x^2 + 3*x + 1]
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> L = NumberField(t**Integer(3) - Integer(3)*t + Integer(1), 'c')
>>> [k[Integer(1)] for k in L.subfields()]
[Ring morphism:
  From: Number Field in c0 with defining polynomial t
  To:   Number Field in c with defining polynomial t^3 - 3*t + 1
  Defn: 0 |--> 0,
 Ring morphism:
  From: Number Field in c1 with defining polynomial t^3 - 3*t + 1
  To:   Number Field in c with defining polynomial t^3 - 3*t + 1
  Defn: c1 |--> c]
>>> len(L.subfields(Integer(2)))
0
>>> len(L.subfields(Integer(1)))
1

sage.rings.number_field.number_field.NumberField_absolute_v1(poly, name, latex_name, canonical_embedding=None)[source]¶

Used for unpickling old pickles.

EXAMPLES:

Sage

sage: from sage.rings.number_field.number_field import NumberField_absolute_v1
sage: R.<x> = QQ[]
sage: NumberField_absolute_v1(x^2 + 1, 'i', 'i')
Number Field in i with defining polynomial x^2 + 1

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field import NumberField_absolute_v1
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> NumberField_absolute_v1(x**Integer(2) + Integer(1), 'i', 'i')
Number Field in i with defining polynomial x^2 + 1

class sage.rings.number_field.number_field.NumberField_cyclotomic(n, names, embedding=None, assume_disc_small=False, maximize_at_primes=None)[source]¶

Bases: NumberField_absolute, NumberField_cyclotomic

Create a cyclotomic extension of the rational field.

The command CyclotomicField(n) creates the \(n\)-th cyclotomic field, obtained by adjoining an \(n\)-th root of unity to the rational field.

EXAMPLES:

Sage

sage: CyclotomicField(3)
Cyclotomic Field of order 3 and degree 2
sage: CyclotomicField(18)
Cyclotomic Field of order 18 and degree 6
sage: z = CyclotomicField(6).gen(); z
zeta6
sage: z^3
-1
sage: (1+z)^3
6*zeta6 - 3

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(3))
Cyclotomic Field of order 3 and degree 2
>>> CyclotomicField(Integer(18))
Cyclotomic Field of order 18 and degree 6
>>> z = CyclotomicField(Integer(6)).gen(); z
zeta6
>>> z**Integer(3)
-1
>>> (Integer(1)+z)**Integer(3)
6*zeta6 - 3

Sage

sage: K = CyclotomicField(197)
sage: loads(K.dumps()) == K
True
sage: loads((z^2).dumps()) == z^2
True

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(197))
>>> loads(K.dumps()) == K
True
>>> loads((z**Integer(2)).dumps()) == z**Integer(2)
True

Sage

sage: cf12 = CyclotomicField(12)
sage: z12 = cf12.0
sage: cf6 = CyclotomicField(6)
sage: z6 = cf6.0
sage: FF = Frac(cf12['x'])
sage: x = FF.0
sage: z6*x^3/(z6 + x)
zeta12^2*x^3/(x + zeta12^2)

Python

>>> from sage.all import *
>>> cf12 = CyclotomicField(Integer(12))
>>> z12 = cf12.gen(0)
>>> cf6 = CyclotomicField(Integer(6))
>>> z6 = cf6.gen(0)
>>> FF = Frac(cf12['x'])
>>> x = FF.gen(0)
>>> z6*x**Integer(3)/(z6 + x)
zeta12^2*x^3/(x + zeta12^2)

Sage

sage: cf6 = CyclotomicField(6); z6 = cf6.gen(0)
sage: cf3 = CyclotomicField(3); z3 = cf3.gen(0)
sage: cf3(z6)
zeta3 + 1
sage: cf6(z3)
zeta6 - 1
sage: type(cf6(z3))
<class 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic'>
sage: cf1 = CyclotomicField(1); z1 = cf1.0
sage: cf3(z1)
1
sage: type(cf3(z1))
<class 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic'>

Python

>>> from sage.all import *
>>> cf6 = CyclotomicField(Integer(6)); z6 = cf6.gen(Integer(0))
>>> cf3 = CyclotomicField(Integer(3)); z3 = cf3.gen(Integer(0))
>>> cf3(z6)
zeta3 + 1
>>> cf6(z3)
zeta6 - 1
>>> type(cf6(z3))
<class 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic'>
>>> cf1 = CyclotomicField(Integer(1)); z1 = cf1.gen(0)
>>> cf3(z1)
1
>>> type(cf3(z1))
<class 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic'>

complex_embedding(prec=53)[source]¶

Return the embedding of this cyclotomic field into the approximate complex field with precision prec obtained by sending the generator \(\zeta\) of self to exp(2*pi*i/n), where \(n\) is the multiplicative order of \(\zeta\).

EXAMPLES:

Sage

sage: C = CyclotomicField(4)
sage: C.complex_embedding()
Ring morphism:
  From: Cyclotomic Field of order 4 and degree 2
  To:   Complex Field with 53 bits of precision
  Defn: zeta4 |--> 6.12323399573677e-17 + 1.00000000000000*I

Python

>>> from sage.all import *
>>> C = CyclotomicField(Integer(4))
>>> C.complex_embedding()
Ring morphism:
  From: Cyclotomic Field of order 4 and degree 2
  To:   Complex Field with 53 bits of precision
  Defn: zeta4 |--> 6.12323399573677e-17 + 1.00000000000000*I

Note in the example above that the way zeta is computed (using sine and cosine in MPFR) means that only the prec bits of the number after the decimal point are valid.

Sage

sage: K = CyclotomicField(3)
sage: phi = K.complex_embedding(10)
sage: phi(K.0)
-0.50 + 0.87*I
sage: phi(K.0^3)
1.0
sage: phi(K.0^3 - 1)
0.00
sage: phi(K.0^3 + 7)
8.0

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(3))
>>> phi = K.complex_embedding(Integer(10))
>>> phi(K.gen(0))
-0.50 + 0.87*I
>>> phi(K.gen(0)**Integer(3))
1.0
>>> phi(K.gen(0)**Integer(3) - Integer(1))
0.00
>>> phi(K.gen(0)**Integer(3) + Integer(7))
8.0

complex_embeddings(prec=53)[source]¶

Return all embeddings of this cyclotomic field into the approximate complex field with precision prec.

If you want 53-bit double precision, which is faster but less reliable, then do self.embeddings(CDF).

EXAMPLES:

Sage

sage: CyclotomicField(5).complex_embeddings()
[
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> 0.309016994374947 + 0.951056516295154*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> -0.809016994374947 + 0.587785252292473*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> -0.809016994374947 - 0.587785252292473*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> 0.309016994374947 - 0.951056516295154*I
]

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(5)).complex_embeddings()
[
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> 0.309016994374947 + 0.951056516295154*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> -0.809016994374947 + 0.587785252292473*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> -0.809016994374947 - 0.587785252292473*I,
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> 0.309016994374947 - 0.951056516295154*I
]

construction()[source]¶

Return data defining a functorial construction of self.

EXAMPLES:

Sage

sage: F, R = CyclotomicField(5).construction()
sage: R
Rational Field
sage: F.polys
[x^4 + x^3 + x^2 + x + 1]
sage: F.names
['zeta5']
sage: F.embeddings
[0.309016994374948? + 0.951056516295154?*I]
sage: F.structures
[None]

Python

>>> from sage.all import *
>>> F, R = CyclotomicField(Integer(5)).construction()
>>> R
Rational Field
>>> F.polys
[x^4 + x^3 + x^2 + x + 1]
>>> F.names
['zeta5']
>>> F.embeddings
[0.309016994374948? + 0.951056516295154?*I]
>>> F.structures
[None]

different()[source]¶

Return the different ideal of the cyclotomic field self.

EXAMPLES:

Sage

sage: C20 = CyclotomicField(20)
sage: C20.different()
Fractional ideal (10, 2*zeta20^6 - 4*zeta20^4 - 4*zeta20^2 + 2)
sage: C18 = CyclotomicField(18)
sage: D = C18.different().norm()
sage: D == C18.discriminant().abs()
True

Python

>>> from sage.all import *
>>> C20 = CyclotomicField(Integer(20))
>>> C20.different()
Fractional ideal (10, 2*zeta20^6 - 4*zeta20^4 - 4*zeta20^2 + 2)
>>> C18 = CyclotomicField(Integer(18))
>>> D = C18.different().norm()
>>> D == C18.discriminant().abs()
True

discriminant(v=None)[source]¶

Return the discriminant of the ring of integers of the cyclotomic field self, or if v is specified, the determinant of the trace pairing on the elements of the list v.

Uses the formula for the discriminant of a prime power cyclotomic field and Hilbert Theorem 88 on the discriminant of composita.

INPUT:

v – (optional) list of elements of this number field

OUTPUT: integer if v is omitted, and Rational otherwise

EXAMPLES:

Sage

sage: CyclotomicField(20).discriminant()
4000000
sage: CyclotomicField(18).discriminant()
-19683

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(20)).discriminant()
4000000
>>> CyclotomicField(Integer(18)).discriminant()
-19683

embeddings(K)[source]¶

Compute all field embeddings of this field into the field \(K\).

INPUT:

K – a field

EXAMPLES:

Sage

sage: CyclotomicField(5).embeddings(ComplexField(53))[1]
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> -0.809016994374947 + 0.587785252292473*I
sage: CyclotomicField(5).embeddings(Qp(11, 4, print_mode='digits'))[1]      # needs sage.rings.padics
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   11-adic Field with capped relative precision 4
  Defn: zeta5 |--> ...1525

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(5)).embeddings(ComplexField(Integer(53)))[Integer(1)]
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Field with 53 bits of precision
  Defn: zeta5 |--> -0.809016994374947 + 0.587785252292473*I
>>> CyclotomicField(Integer(5)).embeddings(Qp(Integer(11), Integer(4), print_mode='digits'))[Integer(1)]      # needs sage.rings.padics
Ring morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   11-adic Field with capped relative precision 4
  Defn: zeta5 |--> ...1525

is_abelian()[source]¶

Return True since all cyclotomic fields are automatically abelian.

EXAMPLES:

Sage

sage: CyclotomicField(29).is_abelian()
True

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(29)).is_abelian()
True

is_galois()[source]¶

Return True since all cyclotomic fields are automatically Galois.

EXAMPLES:

Sage

sage: CyclotomicField(29).is_galois()
True

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(29)).is_galois()
True

is_isomorphic(other)[source]¶

Return True if the cyclotomic field self is isomorphic as a number field to other.

EXAMPLES:

Sage

sage: CyclotomicField(11).is_isomorphic(CyclotomicField(22))
True
sage: CyclotomicField(11).is_isomorphic(CyclotomicField(23))
False
sage: x = polygen(QQ, 'x')
sage: CyclotomicField(3).is_isomorphic(NumberField(x^2 + x + 1, 'a'))
True
sage: CyclotomicField(18).is_isomorphic(CyclotomicField(9))
True
sage: CyclotomicField(10).is_isomorphic(NumberField(x^4 - x^3 + x^2 - x + 1, 'b'))
True

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(11)).is_isomorphic(CyclotomicField(Integer(22)))
True
>>> CyclotomicField(Integer(11)).is_isomorphic(CyclotomicField(Integer(23)))
False
>>> x = polygen(QQ, 'x')
>>> CyclotomicField(Integer(3)).is_isomorphic(NumberField(x**Integer(2) + x + Integer(1), 'a'))
True
>>> CyclotomicField(Integer(18)).is_isomorphic(CyclotomicField(Integer(9)))
True
>>> CyclotomicField(Integer(10)).is_isomorphic(NumberField(x**Integer(4) - x**Integer(3) + x**Integer(2) - x + Integer(1), 'b'))
True

Check Issue #14300:

Sage

sage: K = CyclotomicField(4)
sage: N = K.extension(x^2 - 5, 'z')
sage: K.is_isomorphic(N)
False
sage: K.is_isomorphic(CyclotomicField(8))
False

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(4))
>>> N = K.extension(x**Integer(2) - Integer(5), 'z')
>>> K.is_isomorphic(N)
False
>>> K.is_isomorphic(CyclotomicField(Integer(8)))
False

next_split_prime(p=2)[source]¶

Return the next prime integer \(p\) that splits completely in this cyclotomic field (and does not ramify).

EXAMPLES:

Sage

sage: K.<z> = CyclotomicField(3)
sage: K.next_split_prime(7)
13

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> K.next_split_prime(Integer(7))
13

number_of_roots_of_unity()[source]¶

Return number of roots of unity in this cyclotomic field.

EXAMPLES:

Sage

sage: K.<a> = CyclotomicField(21)
sage: K.number_of_roots_of_unity()
42

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(21), names=('a',)); (a,) = K._first_ngens(1)
>>> K.number_of_roots_of_unity()
42

real_embeddings(prec=53)[source]¶

Return all embeddings of this cyclotomic field into the approximate real field with precision prec.

Mostly, of course, there are no such embeddings.

EXAMPLES:

Sage

sage: len(CyclotomicField(4).real_embeddings())
0
sage: CyclotomicField(2).real_embeddings()
[
Ring morphism:
  From: Cyclotomic Field of order 2 and degree 1
  To:   Real Field with 53 bits of precision
  Defn: -1 |--> -1.00000000000000
]

Python

>>> from sage.all import *
>>> len(CyclotomicField(Integer(4)).real_embeddings())
0
>>> CyclotomicField(Integer(2)).real_embeddings()
[
Ring morphism:
  From: Cyclotomic Field of order 2 and degree 1
  To:   Real Field with 53 bits of precision
  Defn: -1 |--> -1.00000000000000
]

roots_of_unity()[source]¶

Return all the roots of unity in this cyclotomic field, primitive or not.

EXAMPLES:

Sage

sage: K.<a> = CyclotomicField(3)
sage: zs = K.roots_of_unity(); zs
[1, a, -a - 1, -1, -a, a + 1]
sage: [z**K.number_of_roots_of_unity() for z in zs]
[1, 1, 1, 1, 1, 1]

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> zs = K.roots_of_unity(); zs
[1, a, -a - 1, -1, -a, a + 1]
>>> [z**K.number_of_roots_of_unity() for z in zs]
[1, 1, 1, 1, 1, 1]

signature()[source]¶

Return \((r_1, r_2)\), where \(r_1\) and \(r_2\) are the number of real embeddings and pairs of complex embeddings of this cyclotomic field, respectively.

Trivial since, apart from \(\QQ\), cyclotomic fields are totally complex.

EXAMPLES:

Sage

sage: CyclotomicField(5).signature()
(0, 2)
sage: CyclotomicField(2).signature()
(1, 0)

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(5)).signature()
(0, 2)
>>> CyclotomicField(Integer(2)).signature()
(1, 0)

zeta(n=None, all=False)[source]¶

Return an element of multiplicative order \(n\) in this cyclotomic field.

If there is no such element, raise a ValueError.

INPUT:

n – integer (default: None, returns element of maximal order)
all – boolean (default: False); whether to return a list of all primitive \(n\)-th roots of unity

OUTPUT: root of unity or list

EXAMPLES:

Sage

sage: k = CyclotomicField(4)
sage: k.zeta()
zeta4
sage: k.zeta(2)
-1
sage: k.zeta().multiplicative_order()
4

Python

>>> from sage.all import *
>>> k = CyclotomicField(Integer(4))
>>> k.zeta()
zeta4
>>> k.zeta(Integer(2))
-1
>>> k.zeta().multiplicative_order()
4

Sage

sage: k = CyclotomicField(21)
sage: k.zeta().multiplicative_order()
42
sage: k.zeta(21).multiplicative_order()
21
sage: k.zeta(7).multiplicative_order()
7
sage: k.zeta(6).multiplicative_order()
6
sage: k.zeta(84)
Traceback (most recent call last):
...
ValueError: 84 does not divide order of generator (42)

Python

>>> from sage.all import *
>>> k = CyclotomicField(Integer(21))
>>> k.zeta().multiplicative_order()
42
>>> k.zeta(Integer(21)).multiplicative_order()
21
>>> k.zeta(Integer(7)).multiplicative_order()
7
>>> k.zeta(Integer(6)).multiplicative_order()
6
>>> k.zeta(Integer(84))
Traceback (most recent call last):
...
ValueError: 84 does not divide order of generator (42)

Sage

sage: K.<a> = CyclotomicField(7)
sage: K.zeta(all=True)
[-a^4, -a^5, a^5 + a^4 + a^3 + a^2 + a + 1, -a, -a^2, -a^3]
sage: K.zeta(14, all=True)
[-a^4, -a^5, a^5 + a^4 + a^3 + a^2 + a + 1, -a, -a^2, -a^3]
sage: K.zeta(2, all=True)
[-1]
sage: K.<a> = CyclotomicField(10)
sage: K.zeta(20, all=True)
Traceback (most recent call last):
...
ValueError: 20 does not divide order of generator (10)

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> K.zeta(all=True)
[-a^4, -a^5, a^5 + a^4 + a^3 + a^2 + a + 1, -a, -a^2, -a^3]
>>> K.zeta(Integer(14), all=True)
[-a^4, -a^5, a^5 + a^4 + a^3 + a^2 + a + 1, -a, -a^2, -a^3]
>>> K.zeta(Integer(2), all=True)
[-1]
>>> K = CyclotomicField(Integer(10), names=('a',)); (a,) = K._first_ngens(1)
>>> K.zeta(Integer(20), all=True)
Traceback (most recent call last):
...
ValueError: 20 does not divide order of generator (10)

Sage

sage: K.<a> = CyclotomicField(5)
sage: K.zeta(4)
Traceback (most recent call last):
...
ValueError: 4 does not divide order of generator (10)
sage: v = K.zeta(5, all=True); v
[a, a^2, a^3, -a^3 - a^2 - a - 1]
sage: [b^5 for b in v]
[1, 1, 1, 1]

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> K.zeta(Integer(4))
Traceback (most recent call last):
...
ValueError: 4 does not divide order of generator (10)
>>> v = K.zeta(Integer(5), all=True); v
[a, a^2, a^3, -a^3 - a^2 - a - 1]
>>> [b**Integer(5) for b in v]
[1, 1, 1, 1]

zeta_order()[source]¶

Return the order of the maximal root of unity contained in this cyclotomic field.

EXAMPLES:

Sage

sage: CyclotomicField(1).zeta_order()
2
sage: CyclotomicField(4).zeta_order()
4
sage: CyclotomicField(5).zeta_order()
10
sage: CyclotomicField(5)._n()
5
sage: CyclotomicField(389).zeta_order()
778

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(1)).zeta_order()
2
>>> CyclotomicField(Integer(4)).zeta_order()
4
>>> CyclotomicField(Integer(5)).zeta_order()
10
>>> CyclotomicField(Integer(5))._n()
5
>>> CyclotomicField(Integer(389)).zeta_order()
778

sage.rings.number_field.number_field.NumberField_cyclotomic_v1(zeta_order, name, canonical_embedding=None)[source]¶

Used for unpickling old pickles.

EXAMPLES:

Sage

sage: from sage.rings.number_field.number_field import NumberField_cyclotomic_v1
sage: NumberField_cyclotomic_v1(5,'a')
Cyclotomic Field of order 5 and degree 4
sage: NumberField_cyclotomic_v1(5,'a').variable_name()
'a'

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field import NumberField_cyclotomic_v1
>>> NumberField_cyclotomic_v1(Integer(5),'a')
Cyclotomic Field of order 5 and degree 4
>>> NumberField_cyclotomic_v1(Integer(5),'a').variable_name()
'a'

class sage.rings.number_field.number_field.NumberField_generic(polynomial, name, latex_name, check=True, embedding=None, category=None, assume_disc_small=False, maximize_at_primes=None, structure=None)[source]¶

Bases: WithEqualityById, NumberField

Generic class for number fields defined by an irreducible polynomial over \(\QQ\).

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2); K
Number Field in a with defining polynomial x^3 - 2
sage: TestSuite(K).run()

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^3 - 2
>>> TestSuite(K).run()

S_class_group(S, proof=None, names='c')[source]¶

Return the S-class group of this number field over its base field.

INPUT:

S – set of primes of the base field
proof – if False, assume the GRH in computing the class group. Default is True. Call number_field_proof to change this default globally.
names – names of the generators of this class group

OUTPUT: the S-class group of this number field

EXAMPLES:

A well known example:

Sage

sage: K.<a> = QuadraticField(-5)
sage: K.S_class_group([])
S-class group of order 2 with structure C2 of Number Field in a
 with defining polynomial x^2 + 5 with a = 2.236067977499790?*I

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> K.S_class_group([])
S-class group of order 2 with structure C2 of Number Field in a
 with defining polynomial x^2 + 5 with a = 2.236067977499790?*I

When we include the prime \((2, a+1)\), the S-class group becomes trivial:

Sage

sage: K.S_class_group([K.ideal(2, a + 1)])
S-class group of order 1 of Number Field in a
 with defining polynomial x^2 + 5 with a = 2.236067977499790?*I

Python

>>> from sage.all import *
>>> K.S_class_group([K.ideal(Integer(2), a + Integer(1))])
S-class group of order 1 of Number Field in a
 with defining polynomial x^2 + 5 with a = 2.236067977499790?*I

S_unit_group(proof=None, S=None)[source]¶

Return the \(S\)-unit group (including torsion) of this number field.

ALGORITHM: Uses PARI’s pari:bnfsunit command.

INPUT:

proof – boolean (default: True); flag passed to PARI
S – list or tuple of prime ideals, or an ideal, or a single ideal or element from which an ideal can be constructed, in which case the support is used. If None, the global unit group is constructed; otherwise, the \(S\)-unit group is constructed.

Note

The group is cached.

EXAMPLES:

Sage

sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3)
sage: U = K.S_unit_group(S=a); U
S-unit group with structure C10 x Z x Z x Z of
 Number Field in a with defining polynomial x^4 - 10*x^3 + 100*x^2 - 375*x + 1375
 with S = (Fractional ideal (5, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5),
           Fractional ideal (11, 1/275*a^3 + 4/55*a^2 - 5/11*a + 9))
sage: U.gens()
(u0, u1, u2, u3)
sage: U.gens_values()  # random
[-1/275*a^3 + 7/55*a^2 - 6/11*a + 4, 1/275*a^3 + 4/55*a^2 - 5/11*a + 3,
 1/275*a^3 + 4/55*a^2 - 5/11*a + 5, -14/275*a^3 + 21/55*a^2 - 29/11*a + 6]
sage: U.invariants()
(10, 0, 0, 0)
sage: [u.multiplicative_order() for u in U.gens()]
[10, +Infinity, +Infinity, +Infinity]
sage: U.primes()
(Fractional ideal (5, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5),
 Fractional ideal (11, 1/275*a^3 + 4/55*a^2 - 5/11*a + 9))

Python

>>> from sage.all import *
>>> x = polygen(QQ)
>>> K = NumberField(x**Integer(4) - Integer(10)*x**Integer(3) + Integer(20)*Integer(5)*x**Integer(2) - Integer(15)*Integer(5)**Integer(2)*x + Integer(11)*Integer(5)**Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> U = K.S_unit_group(S=a); U
S-unit group with structure C10 x Z x Z x Z of
 Number Field in a with defining polynomial x^4 - 10*x^3 + 100*x^2 - 375*x + 1375
 with S = (Fractional ideal (5, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5),
           Fractional ideal (11, 1/275*a^3 + 4/55*a^2 - 5/11*a + 9))
>>> U.gens()
(u0, u1, u2, u3)
>>> U.gens_values()  # random
[-1/275*a^3 + 7/55*a^2 - 6/11*a + 4, 1/275*a^3 + 4/55*a^2 - 5/11*a + 3,
 1/275*a^3 + 4/55*a^2 - 5/11*a + 5, -14/275*a^3 + 21/55*a^2 - 29/11*a + 6]
>>> U.invariants()
(10, 0, 0, 0)
>>> [u.multiplicative_order() for u in U.gens()]
[10, +Infinity, +Infinity, +Infinity]
>>> U.primes()
(Fractional ideal (5, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5),
 Fractional ideal (11, 1/275*a^3 + 4/55*a^2 - 5/11*a + 9))

With the default value of \(S\), the S-unit group is the same as the global unit group:

Sage

sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^3 + 3)
sage: U = K.unit_group(proof=False)
sage: U.is_isomorphic(K.S_unit_group(proof=False))
True

Python

>>> from sage.all import *
>>> x = polygen(QQ)
>>> K = NumberField(x**Integer(3) + Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> U = K.unit_group(proof=False)
>>> U.is_isomorphic(K.S_unit_group(proof=False))
True

The value of \(S\) may be specified as a list of prime ideals, or an ideal, or an element of the field:

Sage

sage: K.<a> = NumberField(x^3 + 3)
sage: U = K.S_unit_group(proof=False, S=K.ideal(6).prime_factors()); U
S-unit group with structure C2 x Z x Z x Z x Z
 of Number Field in a with defining polynomial x^3 + 3
 with S = (Fractional ideal (-a^2 + a - 1),
           Fractional ideal (a + 1),
           Fractional ideal (a))
sage: K.<a> = NumberField(x^3 + 3)
sage: U = K.S_unit_group(proof=False, S=K.ideal(6)); U
S-unit group with structure C2 x Z x Z x Z x Z
 of Number Field in a with defining polynomial x^3 + 3
 with S = (Fractional ideal (-a^2 + a - 1),
           Fractional ideal (a + 1),
           Fractional ideal (a))
sage: K.<a> = NumberField(x^3 + 3)
sage: U = K.S_unit_group(proof=False, S=6); U
S-unit group with structure C2 x Z x Z x Z x Z
 of Number Field in a with defining polynomial x^3 + 3
 with S = (Fractional ideal (-a^2 + a - 1),
           Fractional ideal (a + 1),
           Fractional ideal (a))
sage: U.primes()
(Fractional ideal (-a^2 + a - 1),
 Fractional ideal (a + 1),
 Fractional ideal (a))
sage: U.gens()
(u0, u1, u2, u3, u4)
sage: U.gens_values()
[-1, a^2 - 2, -a^2 + a - 1, a + 1, a]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) + Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> U = K.S_unit_group(proof=False, S=K.ideal(Integer(6)).prime_factors()); U
S-unit group with structure C2 x Z x Z x Z x Z
 of Number Field in a with defining polynomial x^3 + 3
 with S = (Fractional ideal (-a^2 + a - 1),
           Fractional ideal (a + 1),
           Fractional ideal (a))
>>> K = NumberField(x**Integer(3) + Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> U = K.S_unit_group(proof=False, S=K.ideal(Integer(6))); U
S-unit group with structure C2 x Z x Z x Z x Z
 of Number Field in a with defining polynomial x^3 + 3
 with S = (Fractional ideal (-a^2 + a - 1),
           Fractional ideal (a + 1),
           Fractional ideal (a))
>>> K = NumberField(x**Integer(3) + Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> U = K.S_unit_group(proof=False, S=Integer(6)); U
S-unit group with structure C2 x Z x Z x Z x Z
 of Number Field in a with defining polynomial x^3 + 3
 with S = (Fractional ideal (-a^2 + a - 1),
           Fractional ideal (a + 1),
           Fractional ideal (a))
>>> U.primes()
(Fractional ideal (-a^2 + a - 1),
 Fractional ideal (a + 1),
 Fractional ideal (a))
>>> U.gens()
(u0, u1, u2, u3, u4)
>>> U.gens_values()
[-1, a^2 - 2, -a^2 + a - 1, a + 1, a]

The exp and log methods can be used to create \(S\)-units from sequences of exponents, and recover the exponents:

Sage

sage: U.gens_orders()
(2, 0, 0, 0, 0)
sage: u = U.exp((3,1,4,1,5)); u
-6*a^2 + 18*a - 54
sage: u.norm().factor()
-1 * 2^9 * 3^5
sage: U.log(u)
(1, 1, 4, 1, 5)

Python

>>> from sage.all import *
>>> U.gens_orders()
(2, 0, 0, 0, 0)
>>> u = U.exp((Integer(3),Integer(1),Integer(4),Integer(1),Integer(5))); u
-6*a^2 + 18*a - 54
>>> u.norm().factor()
-1 * 2^9 * 3^5
>>> U.log(u)
(1, 1, 4, 1, 5)

S_unit_solutions(S=[], prec=106, include_exponents=False, include_bound=False, proof=None)[source]¶

Return all solutions to the \(S\)-unit equation \(x + y = 1\) over self.

INPUT:

S – list of finite primes in this number field
prec – precision used for computations in real, complex, and \(p\)-adic fields (default: 106)
include_exponents – whether to include the exponent vectors in the returned value (default: True)
include_bound – whether to return the final computed bound (default: False)
proof – if False, assume the GRH in computing the class group; default is True

OUTPUT:

A list \([(A_1, B_1, x_1, y_1), (A_2, B_2, x_2, y_2), \dots, (A_n, B_n, x_n, y_n)]\) of tuples such that:

The first two entries are tuples \(A_i = (a_0, a_1, \dots, a_t)\) and \(B_i = (b_0, b_1, \dots, b_t)\) of exponents. These will be omitted if include_exponents is False.
The last two entries are \(S\)-units \(x_i\) and \(y_i\) in self with \(x_i + y_i = 1\).
If the default generators for the \(S\)-units of self are \((\rho_0, \rho_1, \dots, \rho_t)`\), then these satisfy \(x_i = \prod(\rho_i)^{(a_i)}\) and \(y_i = \prod(\rho_i)^{(b_i)}\).

If include_bound is True, will return a pair (sols, bound) where sols is as above and bound is the bound used for the entries in the exponent vectors.

EXAMPLES:

Sage

sage: # needs sage.rings.padics
sage: x = polygen(QQ, 'x')
sage: K.<xi> = NumberField(x^2 + x + 1)
sage: S = K.primes_above(3)
sage: K.S_unit_solutions(S)  # random, due to ordering
[(xi + 2, -xi - 1), (1/3*xi + 2/3, -1/3*xi + 1/3), (-xi, xi + 1), (-xi + 1, xi)]

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + x + Integer(1), names=('xi',)); (xi,) = K._first_ngens(1)
>>> S = K.primes_above(Integer(3))
>>> K.S_unit_solutions(S)  # random, due to ordering
[(xi + 2, -xi - 1), (1/3*xi + 2/3, -1/3*xi + 1/3), (-xi, xi + 1), (-xi + 1, xi)]

You can get the exponent vectors:

Sage

sage: # needs sage.rings.padics
sage: K.S_unit_solutions(S, include_exponents=True)  # random, due to ordering
[((2, 1), (4, 0), xi + 2, -xi - 1),
 ((5, -1), (4, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
 ((5, 0), (1, 0), -xi, xi + 1),
 ((1, 1), (2, 0), -xi + 1, xi)]

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> K.S_unit_solutions(S, include_exponents=True)  # random, due to ordering
[((2, 1), (4, 0), xi + 2, -xi - 1),
 ((5, -1), (4, -1), 1/3*xi + 2/3, -1/3*xi + 1/3),
 ((5, 0), (1, 0), -xi, xi + 1),
 ((1, 1), (2, 0), -xi + 1, xi)]

And the computed bound:

Sage

sage: # needs sage.rings.padics
sage: solutions, bound = K.S_unit_solutions(S, prec=100, include_bound=True)
sage: bound
7

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> solutions, bound = K.S_unit_solutions(S, prec=Integer(100), include_bound=True)
>>> bound
7

S_units(S, proof=True)[source]¶

Return a list of generators of the S-units.

INPUT:

S – set of primes of the base field
proof – if False, assume the GRH in computing the class group

OUTPUT: list of generators of the unit group

Note

For more functionality see the function S_unit_group().

EXAMPLES:

Sage

sage: K.<a> = QuadraticField(-3)
sage: K.unit_group()
Unit group with structure C6 of Number Field in a
 with defining polynomial x^2 + 3 with a = 1.732050807568878?*I
sage: K.S_units([])  # random
[1/2*a + 1/2]
sage: K.S_units([])[0].multiplicative_order()
6

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> K.unit_group()
Unit group with structure C6 of Number Field in a
 with defining polynomial x^2 + 3 with a = 1.732050807568878?*I
>>> K.S_units([])  # random
[1/2*a + 1/2]
>>> K.S_units([])[Integer(0)].multiplicative_order()
6

An example in a relative extension (see Issue #8722):

Sage

sage: x = polygen(QQ, 'x')
sage: L.<a,b> = NumberField([x^2 + 1, x^2 - 5])
sage: p = L.ideal((-1/2*b - 1/2)*a + 1/2*b - 1/2)
sage: W = L.S_units([p]); [x.norm() for x in W]
[9, 1, 1]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> L = NumberField([x**Integer(2) + Integer(1), x**Integer(2) - Integer(5)], names=('a', 'b',)); (a, b,) = L._first_ngens(2)
>>> p = L.ideal((-Integer(1)/Integer(2)*b - Integer(1)/Integer(2))*a + Integer(1)/Integer(2)*b - Integer(1)/Integer(2))
>>> W = L.S_units([p]); [x.norm() for x in W]
[9, 1, 1]

Our generators should have the correct parent (Issue #9367):

Sage

sage: _.<x> = QQ[]
sage: L.<alpha> = NumberField(x^3 + x + 1)
sage: p = L.S_units([ L.ideal(7) ])
sage: p[0].parent()
Number Field in alpha with defining polynomial x^3 + x + 1

Python

>>> from sage.all import *
>>> _ = QQ['x']; (x,) = _._first_ngens(1)
>>> L = NumberField(x**Integer(3) + x + Integer(1), names=('alpha',)); (alpha,) = L._first_ngens(1)
>>> p = L.S_units([ L.ideal(Integer(7)) ])
>>> p[Integer(0)].parent()
Number Field in alpha with defining polynomial x^3 + x + 1

absolute_degree()[source]¶

Return the degree of self over \(\QQ\).

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').absolute_degree()
3
sage: NumberField(x + 1, 'a').absolute_degree()
1
sage: NumberField(x^997 + 17*x + 3, 'a', check=False).absolute_degree()
997

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(3) + x**Integer(2) + Integer(997)*x + Integer(1), 'a').absolute_degree()
3
>>> NumberField(x + Integer(1), 'a').absolute_degree()
1
>>> NumberField(x**Integer(997) + Integer(17)*x + Integer(3), 'a', check=False).absolute_degree()
997

absolute_field(names)[source]¶

Return self as an absolute number field.

INPUT:

names – string; name of generator of the absolute field

OUTPUT:

K – this number field (since it is already absolute)

Also, K.structure() returns from_K and to_K, where from_K is an isomorphism from \(K\) to self and to_K is an isomorphism from self to \(K\).

EXAMPLES:

Sage

sage: K = CyclotomicField(5)
sage: K.absolute_field('a')
Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(5))
>>> K.absolute_field('a')
Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1

absolute_polynomial_ntl()[source]¶

Alias for polynomial_ntl(). Mostly for internal use.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + (2/3)*x - 9/17,'a').absolute_polynomial_ntl()
([-27 34 51], 51)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + (Integer(2)/Integer(3))*x - Integer(9)/Integer(17),'a').absolute_polynomial_ntl()
([-27 34 51], 51)

algebraic_closure()[source]¶

Return the algebraic closure of self (which is QQbar).

EXAMPLES:

Sage

sage: K.<i> = QuadraticField(-1)
sage: K.algebraic_closure()
Algebraic Field
sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: K.algebraic_closure()
Algebraic Field
sage: K = CyclotomicField(23)
sage: K.algebraic_closure()
Algebraic Field

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.algebraic_closure()
Algebraic Field
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.algebraic_closure()
Algebraic Field
>>> K = CyclotomicField(Integer(23))
>>> K.algebraic_closure()
Algebraic Field

change_generator(alpha, name=None, names=None)[source]¶

Given the number field self, construct another isomorphic number field \(K\) generated by the element alpha of self, along with isomorphisms from \(K\) to self and from self to \(K\).

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: L.<i> = NumberField(x^2 + 1); L
Number Field in i with defining polynomial x^2 + 1
sage: K, from_K, to_K = L.change_generator(i/2 + 3)
sage: K
Number Field in i0 with defining polynomial x^2 - 6*x + 37/4 with i0 = 1/2*i + 3
sage: from_K
Ring morphism:
  From: Number Field in i0 with defining polynomial x^2 - 6*x + 37/4
        with i0 = 1/2*i + 3
  To:   Number Field in i with defining polynomial x^2 + 1
  Defn: i0 |--> 1/2*i + 3
sage: to_K
Ring morphism:
  From: Number Field in i with defining polynomial x^2 + 1
  To:   Number Field in i0 with defining polynomial x^2 - 6*x + 37/4
        with i0 = 1/2*i + 3
  Defn: i |--> 2*i0 - 6

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> L = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = L._first_ngens(1); L
Number Field in i with defining polynomial x^2 + 1
>>> K, from_K, to_K = L.change_generator(i/Integer(2) + Integer(3))
>>> K
Number Field in i0 with defining polynomial x^2 - 6*x + 37/4 with i0 = 1/2*i + 3
>>> from_K
Ring morphism:
  From: Number Field in i0 with defining polynomial x^2 - 6*x + 37/4
        with i0 = 1/2*i + 3
  To:   Number Field in i with defining polynomial x^2 + 1
  Defn: i0 |--> 1/2*i + 3
>>> to_K
Ring morphism:
  From: Number Field in i with defining polynomial x^2 + 1
  To:   Number Field in i0 with defining polynomial x^2 - 6*x + 37/4
        with i0 = 1/2*i + 3
  Defn: i |--> 2*i0 - 6

We can also do

Sage

sage: K.<c>, from_K, to_K = L.change_generator(i/2 + 3); K
Number Field in c with defining polynomial x^2 - 6*x + 37/4 with c = 1/2*i + 3

Python

>>> from sage.all import *
>>> K, from_K, to_K  = L.change_generator(i/Integer(2) + Integer(3), names=('c',)); (c,) = K._first_ngens(1); K
Number Field in c with defining polynomial x^2 - 6*x + 37/4 with c = 1/2*i + 3

We compute the image of the generator \(\sqrt{-1}\) of \(L\).

Sage

sage: to_K(i)
2*c - 6

Python

>>> from sage.all import *
>>> to_K(i)
2*c - 6

Note that the image is indeed a square root of \(-1\).

Sage

sage: to_K(i)^2
-1
sage: from_K(to_K(i))
i
sage: to_K(from_K(c))
c

Python

>>> from sage.all import *
>>> to_K(i)**Integer(2)
-1
>>> from_K(to_K(i))
i
>>> to_K(from_K(c))
c

characteristic()[source]¶

Return the characteristic of this number field, which is of course 0.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^99 + 2); k
Number Field in a with defining polynomial x^99 + 2
sage: k.characteristic()
0

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(99) + Integer(2), names=('a',)); (a,) = k._first_ngens(1); k
Number Field in a with defining polynomial x^99 + 2
>>> k.characteristic()
0

class_group(proof=None, names='c')[source]¶

Return the class group of the ring of integers of this number field.

INPUT:

proof – if True (default), then compute the class group provably correctly; call number_field_proof() to change this default globally
names – names of the generators of this class group

OUTPUT: the class group of this number field

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 + 23)
sage: G = K.class_group(); G
Class group of order 3 with structure C3 of
 Number Field in a with defining polynomial x^2 + 23
sage: G.0
Fractional ideal class (2, 1/2*a - 1/2)
sage: G.gens()
(Fractional ideal class (2, 1/2*a - 1/2),)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> G = K.class_group(); G
Class group of order 3 with structure C3 of
 Number Field in a with defining polynomial x^2 + 23
>>> G.gen(0)
Fractional ideal class (2, 1/2*a - 1/2)
>>> G.gens()
(Fractional ideal class (2, 1/2*a - 1/2),)

Sage

sage: G.number_field()
Number Field in a with defining polynomial x^2 + 23
sage: G is K.class_group()
True
sage: G is K.class_group(proof=False)
False
sage: G.gens()
(Fractional ideal class (2, 1/2*a - 1/2),)

Python

>>> from sage.all import *
>>> G.number_field()
Number Field in a with defining polynomial x^2 + 23
>>> G is K.class_group()
True
>>> G is K.class_group(proof=False)
False
>>> G.gens()
(Fractional ideal class (2, 1/2*a - 1/2),)

There can be multiple generators:

Sage

sage: k.<a> = NumberField(x^2 + 20072)
sage: G = k.class_group(); G
Class group of order 76 with structure C38 x C2 of
 Number Field in a with defining polynomial x^2 + 20072
sage: G.0 # random
Fractional ideal class (41, a + 10)
sage: G.0^38
Trivial principal fractional ideal class
sage: G.1 # random
Fractional ideal class (2, -1/2*a)
sage: G.1^2
Trivial principal fractional ideal class

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(2) + Integer(20072), names=('a',)); (a,) = k._first_ngens(1)
>>> G = k.class_group(); G
Class group of order 76 with structure C38 x C2 of
 Number Field in a with defining polynomial x^2 + 20072
>>> G.gen(0) # random
Fractional ideal class (41, a + 10)
>>> G.gen(0)**Integer(38)
Trivial principal fractional ideal class
>>> G.gen(1) # random
Fractional ideal class (2, -1/2*a)
>>> G.gen(1)**Integer(2)
Trivial principal fractional ideal class

Class groups of Hecke polynomials tend to be very small:

Sage

sage: # needs sage.modular
sage: f = ModularForms(97, 2).T(2).charpoly()
sage: f.factor()
(x - 3) * (x^3 + 4*x^2 + 3*x - 1) * (x^4 - 3*x^3 - x^2 + 6*x - 1)
sage: [NumberField(g,'a').class_group().order() for g,_ in f.factor()]
[1, 1, 1]

Python

>>> from sage.all import *
>>> # needs sage.modular
>>> f = ModularForms(Integer(97), Integer(2)).T(Integer(2)).charpoly()
>>> f.factor()
(x - 3) * (x^3 + 4*x^2 + 3*x - 1) * (x^4 - 3*x^3 - x^2 + 6*x - 1)
>>> [NumberField(g,'a').class_group().order() for g,_ in f.factor()]
[1, 1, 1]

Note

Unlike in PARI/GP, class group computations in Sage do not by default assume the Generalized Riemann Hypothesis. To do class groups computations not provably correctly you must often pass the flag proof=False to functions or call the function proof.number_field(False). It can easily take 1000s of times longer to do computations with proof=True (the default).

class_number(proof=None)[source]¶

Return the class number of this number field, as an integer.

INPUT:

proof – boolean (default: True, unless you called number_field_proof)

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + 23, 'a').class_number()
3
sage: NumberField(x^2 + 163, 'a').class_number()
1
sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').class_number(proof=False)
1539

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + Integer(23), 'a').class_number()
3
>>> NumberField(x**Integer(2) + Integer(163), 'a').class_number()
1
>>> NumberField(x**Integer(3) + x**Integer(2) + Integer(997)*x + Integer(1), 'a').class_number(proof=False)
1539

completely_split_primes(B=200)[source]¶

Return a list of rational primes which split completely in the number field \(K\).

INPUT:

B – positive integer bound (default: 200)

OUTPUT: list of all primes \(p < B\) which split completely in K

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<xi> = NumberField(x^3 - 3*x + 1)
sage: K.completely_split_primes(100)
[17, 19, 37, 53, 71, 73, 89]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(3)*x + Integer(1), names=('xi',)); (xi,) = K._first_ngens(1)
>>> K.completely_split_primes(Integer(100))
[17, 19, 37, 53, 71, 73, 89]

completion(p, prec, extras={})[source]¶

Return the completion of self at \(p\) to the specified precision.

Only implemented at archimedean places, and then only if an embedding has been fixed.

EXAMPLES:

Sage

sage: K.<a> = QuadraticField(2)
sage: K.completion(infinity, 100)
Real Field with 100 bits of precision
sage: K.<zeta> = CyclotomicField(12)
sage: K.completion(infinity, 53, extras={'type': 'RDF'})
Complex Double Field
sage: zeta + 1.5                            # implicit test
2.36602540378444 + 0.500000000000000*I

Python

>>> from sage.all import *
>>> K = QuadraticField(Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.completion(infinity, Integer(100))
Real Field with 100 bits of precision
>>> K = CyclotomicField(Integer(12), names=('zeta',)); (zeta,) = K._first_ngens(1)
>>> K.completion(infinity, Integer(53), extras={'type': 'RDF'})
Complex Double Field
>>> zeta + RealNumber('1.5')                            # implicit test
2.36602540378444 + 0.500000000000000*I

complex_conjugation()[source]¶

Return the complex conjugation of self.

This is only well-defined for fields contained in CM fields (i.e. for totally real fields and CM fields). Recall that a CM field is a totally imaginary quadratic extension of a totally real field. For other fields, a ValueError is raised.

EXAMPLES:

Sage

sage: QuadraticField(-1, 'I').complex_conjugation()
Ring endomorphism of
 Number Field in I with defining polynomial x^2 + 1 with I = 1*I
  Defn: I |--> -I
sage: CyclotomicField(8).complex_conjugation()
Ring endomorphism of Cyclotomic Field of order 8 and degree 4
  Defn: zeta8 |--> -zeta8^3
sage: QuadraticField(5, 'a').complex_conjugation()
Identity endomorphism of Number Field in a with defining
 polynomial x^2 - 5 with a = 2.236067977499790?
sage: x = polygen(QQ, 'x')
sage: F = NumberField(x^4 + x^3 - 3*x^2 - x + 1, 'a')
sage: F.is_totally_real()
True
sage: F.complex_conjugation()
Identity endomorphism of Number Field in a with defining
 polynomial x^4 + x^3 - 3*x^2 - x + 1
sage: F.<b> = NumberField(x^2 - 2)
sage: F.extension(x^2 + 1, 'a').complex_conjugation()
Relative number field endomorphism of Number Field in a
 with defining polynomial x^2 + 1 over its base field
  Defn: a |--> -a
        b |--> b
sage: F2.<b> = NumberField(x^2 + 2)
sage: K2.<a> = F2.extension(x^2 + 1)
sage: cc = K2.complex_conjugation()
sage: cc(a)
-a
sage: cc(b)
-b

Python

>>> from sage.all import *
>>> QuadraticField(-Integer(1), 'I').complex_conjugation()
Ring endomorphism of
 Number Field in I with defining polynomial x^2 + 1 with I = 1*I
  Defn: I |--> -I
>>> CyclotomicField(Integer(8)).complex_conjugation()
Ring endomorphism of Cyclotomic Field of order 8 and degree 4
  Defn: zeta8 |--> -zeta8^3
>>> QuadraticField(Integer(5), 'a').complex_conjugation()
Identity endomorphism of Number Field in a with defining
 polynomial x^2 - 5 with a = 2.236067977499790?
>>> x = polygen(QQ, 'x')
>>> F = NumberField(x**Integer(4) + x**Integer(3) - Integer(3)*x**Integer(2) - x + Integer(1), 'a')
>>> F.is_totally_real()
True
>>> F.complex_conjugation()
Identity endomorphism of Number Field in a with defining
 polynomial x^4 + x^3 - 3*x^2 - x + 1
>>> F = NumberField(x**Integer(2) - Integer(2), names=('b',)); (b,) = F._first_ngens(1)
>>> F.extension(x**Integer(2) + Integer(1), 'a').complex_conjugation()
Relative number field endomorphism of Number Field in a
 with defining polynomial x^2 + 1 over its base field
  Defn: a |--> -a
        b |--> b
>>> F2 = NumberField(x**Integer(2) + Integer(2), names=('b',)); (b,) = F2._first_ngens(1)
>>> K2 = F2.extension(x**Integer(2) + Integer(1), names=('a',)); (a,) = K2._first_ngens(1)
>>> cc = K2.complex_conjugation()
>>> cc(a)
-a
>>> cc(b)
-b

complex_embeddings(prec=53)[source]¶

Return all homomorphisms of this number field into the approximate complex field with precision prec.

This always embeds into an MPFR based complex field. If you want embeddings into the 53-bit double precision, which is faster, use self.embeddings(CDF).

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^5 + x + 17)
sage: v = k.complex_embeddings()
sage: ls = [phi(k.0^2) for phi in v]; ls  # random order
[2.97572074038...,
 -2.40889943716 + 1.90254105304*I,
 -2.40889943716 - 1.90254105304*I,
 0.921039066973 + 3.07553311885*I,
 0.921039066973 - 3.07553311885*I]
sage: K.<a> = NumberField(x^3 + 2)
sage: ls = K.complex_embeddings(); ls  # random order
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Complex Double Field
  Defn: a |--> -1.25992104989...,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Complex Double Field
  Defn: a |--> 0.629960524947 - 1.09112363597*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Complex Double Field
  Defn: a |--> 0.629960524947 + 1.09112363597*I
]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(5) + x + Integer(17), names=('a',)); (a,) = k._first_ngens(1)
>>> v = k.complex_embeddings()
>>> ls = [phi(k.gen(0)**Integer(2)) for phi in v]; ls  # random order
[2.97572074038...,
 -2.40889943716 + 1.90254105304*I,
 -2.40889943716 - 1.90254105304*I,
 0.921039066973 + 3.07553311885*I,
 0.921039066973 - 3.07553311885*I]
>>> K = NumberField(x**Integer(3) + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> ls = K.complex_embeddings(); ls  # random order
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Complex Double Field
  Defn: a |--> -1.25992104989...,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Complex Double Field
  Defn: a |--> 0.629960524947 - 1.09112363597*I,
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Complex Double Field
  Defn: a |--> 0.629960524947 + 1.09112363597*I
]

composite_fields(other, names=None, both_maps=False, preserve_embedding=True)[source]¶

Return the possible composite number fields formed from self and other.

INPUT:

other – number field
names – generator name for composite fields
both_maps – boolean (default: False)
preserve_embedding – boolean (default: True)

OUTPUT: list of the composite fields, possibly with maps

If both_maps is True, the list consists of quadruples (F, self_into_F, other_into_F, k) such that self_into_F is an embedding of self in F, other_into_F is an embedding of in F, and k is one of the following:

an integer such that F.gen() equals other_into_F(other.gen()) + k*self_into_F(self.gen());
Infinity, in which case F.gen() equals self_into_F(self.gen());
None (when other is a relative number field).

If both self and other have embeddings into an ambient field, then each F will have an embedding with respect to which both self_into_F and other_into_F will be compatible with the ambient embeddings.

If preserve_embedding is True and if self and other both have embeddings into the same ambient field, or into fields which are contained in a common field, only the compositum respecting both embeddings is returned. In all other cases, all possible composite number fields are returned.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^4 - 2)
sage: K.composite_fields(K)
[Number Field in a with defining polynomial x^4 - 2,
 Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(4) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.composite_fields(K)
[Number Field in a with defining polynomial x^4 - 2,
 Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500]

A particular compositum is selected, together with compatible maps into the compositum, if the fields are endowed with a real or complex embedding:

Sage

sage: # needs sage.symbolic
sage: K1 = NumberField(x^4 - 2, 'a', embedding=RR(2^(1/4)))
sage: K2 = NumberField(x^4 - 2, 'a', embedding=RR(-2^(1/4)))
sage: K1.composite_fields(K2)
[Number Field in a with defining polynomial x^4 - 2 with a = 1.189207115002722?]
sage: [F, f, g, k], = K1.composite_fields(K2, both_maps=True); F
Number Field in a with defining polynomial x^4 - 2 with a = 1.189207115002722?
sage: f(K1.0), g(K2.0)
(a, -a)

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> K1 = NumberField(x**Integer(4) - Integer(2), 'a', embedding=RR(Integer(2)**(Integer(1)/Integer(4))))
>>> K2 = NumberField(x**Integer(4) - Integer(2), 'a', embedding=RR(-Integer(2)**(Integer(1)/Integer(4))))
>>> K1.composite_fields(K2)
[Number Field in a with defining polynomial x^4 - 2 with a = 1.189207115002722?]
>>> [F, f, g, k], = K1.composite_fields(K2, both_maps=True); F
Number Field in a with defining polynomial x^4 - 2 with a = 1.189207115002722?
>>> f(K1.gen(0)), g(K2.gen(0))
(a, -a)

With preserve_embedding set to False, the embeddings are ignored:

Sage

sage: K1.composite_fields(K2, preserve_embedding=False)                     # needs sage.symbolic
[Number Field in a with defining polynomial x^4 - 2 with a = 1.189207115002722?,
 Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500]

Python

>>> from sage.all import *
>>> K1.composite_fields(K2, preserve_embedding=False)                     # needs sage.symbolic
[Number Field in a with defining polynomial x^4 - 2 with a = 1.189207115002722?,
 Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500]

Changing the embedding selects a different compositum:

Sage

sage: K3 = NumberField(x^4 - 2, 'a', embedding=CC(2^(1/4)*I))               # needs sage.symbolic
sage: [F, f, g, k], = K1.composite_fields(K3, both_maps=True); F            # needs sage.symbolic
Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500
 with a0 = -2.378414230005443? + 1.189207115002722?*I
sage: f(K1.0), g(K3.0)                                                      # needs sage.symbolic
(1/240*a0^5 - 41/120*a0, 1/120*a0^5 + 19/60*a0)

Python

>>> from sage.all import *
>>> K3 = NumberField(x**Integer(4) - Integer(2), 'a', embedding=CC(Integer(2)**(Integer(1)/Integer(4))*I))               # needs sage.symbolic
>>> [F, f, g, k], = K1.composite_fields(K3, both_maps=True); F            # needs sage.symbolic
Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500
 with a0 = -2.378414230005443? + 1.189207115002722?*I
>>> f(K1.gen(0)), g(K3.gen(0))                                                      # needs sage.symbolic
(1/240*a0^5 - 41/120*a0, 1/120*a0^5 + 19/60*a0)

If no embeddings are specified, the maps into the compositum are chosen arbitrarily:

Sage

sage: Q1.<a> = NumberField(x^4 + 10*x^2 + 1)
sage: Q2.<b> = NumberField(x^4 + 16*x^2 + 4)
sage: Q1.composite_fields(Q2, 'c')
[Number Field in c with defining polynomial
  x^8 + 64*x^6 + 904*x^4 + 3840*x^2 + 3600]
sage: F, Q1_into_F, Q2_into_F, k = Q1.composite_fields(Q2, 'c',
....:                                                  both_maps=True)[0]
sage: Q1_into_F
Ring morphism:
  From: Number Field in a with defining polynomial x^4 + 10*x^2 + 1
  To:   Number Field in c with defining polynomial
        x^8 + 64*x^6 + 904*x^4 + 3840*x^2 + 3600
  Defn: a |--> 19/14400*c^7 + 137/1800*c^5 + 2599/3600*c^3 + 8/15*c

Python

>>> from sage.all import *
>>> Q1 = NumberField(x**Integer(4) + Integer(10)*x**Integer(2) + Integer(1), names=('a',)); (a,) = Q1._first_ngens(1)
>>> Q2 = NumberField(x**Integer(4) + Integer(16)*x**Integer(2) + Integer(4), names=('b',)); (b,) = Q2._first_ngens(1)
>>> Q1.composite_fields(Q2, 'c')
[Number Field in c with defining polynomial
  x^8 + 64*x^6 + 904*x^4 + 3840*x^2 + 3600]
>>> F, Q1_into_F, Q2_into_F, k = Q1.composite_fields(Q2, 'c',
...                                                  both_maps=True)[Integer(0)]
>>> Q1_into_F
Ring morphism:
  From: Number Field in a with defining polynomial x^4 + 10*x^2 + 1
  To:   Number Field in c with defining polynomial
        x^8 + 64*x^6 + 904*x^4 + 3840*x^2 + 3600
  Defn: a |--> 19/14400*c^7 + 137/1800*c^5 + 2599/3600*c^3 + 8/15*c

This is just one of four embeddings of Q1 into F:

Sage

sage: Hom(Q1, F).order()
4

Python

>>> from sage.all import *
>>> Hom(Q1, F).order()
4

Note that even with preserve_embedding=True, this method may fail to recognize that the two number fields have compatible embeddings, and hence return several composite number fields:

Sage

sage: x = polygen(ZZ)
sage: A.<a> = NumberField(x^3 - 7, embedding=CC(-0.95+1.65*I))
sage: r = QQbar.polynomial_root(x^9 - 7, RIF(1.2, 1.3))
sage: B.<a> = NumberField(x^9 - 7, embedding=r)
sage: len(A.composite_fields(B, preserve_embedding=True))
2

Python

>>> from sage.all import *
>>> x = polygen(ZZ)
>>> A = NumberField(x**Integer(3) - Integer(7), embedding=CC(-RealNumber('0.95')+RealNumber('1.65')*I), names=('a',)); (a,) = A._first_ngens(1)
>>> r = QQbar.polynomial_root(x**Integer(9) - Integer(7), RIF(RealNumber('1.2'), RealNumber('1.3')))
>>> B = NumberField(x**Integer(9) - Integer(7), embedding=r, names=('a',)); (a,) = B._first_ngens(1)
>>> len(A.composite_fields(B, preserve_embedding=True))
2

conductor(check_abelian=True)[source]¶

Compute the conductor of the abelian field \(K\). If check_abelian is set to False and the field is not an abelian extension of \(\QQ\), the output is not meaningful.

INPUT:

check_abelian – boolean (default: True); check to see that this is an abelian extension of \(\QQ\)

OUTPUT: integer which is the conductor of the field

EXAMPLES:

Sage

sage: # needs sage.groups
sage: K = CyclotomicField(27)
sage: k = K.subfields(9)[0][0]
sage: k.conductor()
27
sage: x = polygen(QQ, 'x')
sage: K.<t> = NumberField(x^3 + x^2 - 2*x - 1)
sage: K.conductor()
7
sage: K.<t> = NumberField(x^3 + x^2 - 36*x - 4)
sage: K.conductor()
109
sage: K = CyclotomicField(48)
sage: k = K.subfields(16)[0][0]
sage: k.conductor()
48
sage: NumberField(x,'a').conductor()
1
sage: NumberField(x^8 - 8*x^6 + 19*x^4 - 12*x^2 + 1, 'a').conductor()
40
sage: NumberField(x^8 + 7*x^4 + 1, 'a').conductor()
40
sage: NumberField(x^8 - 40*x^6 + 500*x^4 - 2000*x^2 + 50, 'a').conductor()
160

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> K = CyclotomicField(Integer(27))
>>> k = K.subfields(Integer(9))[Integer(0)][Integer(0)]
>>> k.conductor()
27
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x - Integer(1), names=('t',)); (t,) = K._first_ngens(1)
>>> K.conductor()
7
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(36)*x - Integer(4), names=('t',)); (t,) = K._first_ngens(1)
>>> K.conductor()
109
>>> K = CyclotomicField(Integer(48))
>>> k = K.subfields(Integer(16))[Integer(0)][Integer(0)]
>>> k.conductor()
48
>>> NumberField(x,'a').conductor()
1
>>> NumberField(x**Integer(8) - Integer(8)*x**Integer(6) + Integer(19)*x**Integer(4) - Integer(12)*x**Integer(2) + Integer(1), 'a').conductor()
40
>>> NumberField(x**Integer(8) + Integer(7)*x**Integer(4) + Integer(1), 'a').conductor()
40
>>> NumberField(x**Integer(8) - Integer(40)*x**Integer(6) + Integer(500)*x**Integer(4) - Integer(2000)*x**Integer(2) + Integer(50), 'a').conductor()
160

ALGORITHM:

For odd primes, it is easy to compute from the ramification index because the \(p\)-Sylow subgroup is cyclic. For \(p=2\), there are two choices for a given ramification index. They can be distinguished by the parity of the exponent in the discriminant of a 2-adic completion.

construction()[source]¶

Construction of self.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 + x^2 + 1, embedding=CC.gen())
sage: F, R = K.construction()
sage: F
AlgebraicExtensionFunctor
sage: R
Rational Field

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + x**Integer(2) + Integer(1), embedding=CC.gen(), names=('a',)); (a,) = K._first_ngens(1)
>>> F, R = K.construction()
>>> F
AlgebraicExtensionFunctor
>>> R
Rational Field

The construction functor respects distinguished embeddings:

Sage

sage: F(R) is K
True
sage: F.embeddings
[0.2327856159383841? + 0.7925519925154479?*I]

Python

>>> from sage.all import *
>>> F(R) is K
True
>>> F.embeddings
[0.2327856159383841? + 0.7925519925154479?*I]

decomposition_type(p)[source]¶

Return how the given prime of the base field splits in this number field.

INPUT:

p – a prime element or ideal of the base field

OUTPUT:

A list of triples \((e, f, g)\) where

\(e\) is the ramification index,
\(f\) is the residue class degree,
\(g\) is the number of primes above \(p\) with given \(e\) and \(f\)

EXAMPLES:

Sage

sage: R.<x> = ZZ[]
sage: K.<a> = NumberField(x^20 + 3*x^18 + 15*x^16 + 28*x^14 + 237*x^12 + 579*x^10
....:                      + 1114*x^8 + 1470*x^6 + 2304*x^4 + 1296*x^2 + 729)
sage: K.is_galois()                                                         # needs sage.groups
True
sage: K.discriminant().factor()
2^20 * 3^10 * 53^10
sage: K.decomposition_type(2)
[(2, 5, 2)]
sage: K.decomposition_type(3)
[(2, 1, 10)]
sage: K.decomposition_type(53)
[(2, 2, 5)]

Python

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(20) + Integer(3)*x**Integer(18) + Integer(15)*x**Integer(16) + Integer(28)*x**Integer(14) + Integer(237)*x**Integer(12) + Integer(579)*x**Integer(10)
...                      + Integer(1114)*x**Integer(8) + Integer(1470)*x**Integer(6) + Integer(2304)*x**Integer(4) + Integer(1296)*x**Integer(2) + Integer(729), names=('a',)); (a,) = K._first_ngens(1)
>>> K.is_galois()                                                         # needs sage.groups
True
>>> K.discriminant().factor()
2^20 * 3^10 * 53^10
>>> K.decomposition_type(Integer(2))
[(2, 5, 2)]
>>> K.decomposition_type(Integer(3))
[(2, 1, 10)]
>>> K.decomposition_type(Integer(53))
[(2, 2, 5)]

This example is only ramified at 11:

Sage

sage: K.<a> = NumberField(x^24 + 11^2*(90*x^12 - 640*x^8 + 2280*x^6
....:                                   - 512*x^4 + 2432/11*x^2 - 11))
sage: K.discriminant().factor()
-1 * 11^43
sage: K.decomposition_type(11)
[(1, 1, 2), (22, 1, 1)]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(24) + Integer(11)**Integer(2)*(Integer(90)*x**Integer(12) - Integer(640)*x**Integer(8) + Integer(2280)*x**Integer(6)
...                                   - Integer(512)*x**Integer(4) + Integer(2432)/Integer(11)*x**Integer(2) - Integer(11)), names=('a',)); (a,) = K._first_ngens(1)
>>> K.discriminant().factor()
-1 * 11^43
>>> K.decomposition_type(Integer(11))
[(1, 1, 2), (22, 1, 1)]

Computing the decomposition type is feasible even in large degree:

Sage

sage: K.<a> = NumberField(x^144 + 123*x^72 + 321*x^36 + 13*x^18 + 11)
sage: K.discriminant().factor(limit=100000)
2^144 * 3^288 * 7^18 * 11^17 * 31^18 * 157^18 * 2153^18 * 13907^18 * ...
sage: K.decomposition_type(2)
[(2, 4, 3), (2, 12, 2), (2, 36, 1)]
sage: K.decomposition_type(3)
[(9, 3, 2), (9, 10, 1)]
sage: K.decomposition_type(7)
[(1, 18, 1), (1, 90, 1), (2, 1, 6), (2, 3, 4)]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(144) + Integer(123)*x**Integer(72) + Integer(321)*x**Integer(36) + Integer(13)*x**Integer(18) + Integer(11), names=('a',)); (a,) = K._first_ngens(1)
>>> K.discriminant().factor(limit=Integer(100000))
2^144 * 3^288 * 7^18 * 11^17 * 31^18 * 157^18 * 2153^18 * 13907^18 * ...
>>> K.decomposition_type(Integer(2))
[(2, 4, 3), (2, 12, 2), (2, 36, 1)]
>>> K.decomposition_type(Integer(3))
[(9, 3, 2), (9, 10, 1)]
>>> K.decomposition_type(Integer(7))
[(1, 18, 1), (1, 90, 1), (2, 1, 6), (2, 3, 4)]

It also works for relative extensions:

Sage

sage: K.<a> = QuadraticField(-143)
sage: M.<c> = K.extension(x^10 - 6*x^8 + (a + 12)*x^6 + (-7/2*a - 89/2)*x^4
....:                      + (13/2*a - 77/2)*x^2 + 25)

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(143), names=('a',)); (a,) = K._first_ngens(1)
>>> M = K.extension(x**Integer(10) - Integer(6)*x**Integer(8) + (a + Integer(12))*x**Integer(6) + (-Integer(7)/Integer(2)*a - Integer(89)/Integer(2))*x**Integer(4)
...                      + (Integer(13)/Integer(2)*a - Integer(77)/Integer(2))*x**Integer(2) + Integer(25), names=('c',)); (c,) = M._first_ngens(1)

There is a unique prime above \(11\) and above \(13\) in \(K\), each of which is unramified in \(M\):

Sage

sage: M.decomposition_type(11)
[(1, 2, 5)]
sage: P11 = K.primes_above(11)[0]
sage: len(M.primes_above(P11))
5
sage: M.decomposition_type(13)
[(1, 1, 10)]
sage: P13 = K.primes_above(13)[0]
sage: len(M.primes_above(P13))
10

Python

>>> from sage.all import *
>>> M.decomposition_type(Integer(11))
[(1, 2, 5)]
>>> P11 = K.primes_above(Integer(11))[Integer(0)]
>>> len(M.primes_above(P11))
5
>>> M.decomposition_type(Integer(13))
[(1, 1, 10)]
>>> P13 = K.primes_above(Integer(13))[Integer(0)]
>>> len(M.primes_above(P13))
10

There are two primes above \(2\), each of which ramifies in \(M\):

Sage

sage: Q0, Q1 = K.primes_above(2)
sage: M.decomposition_type(Q0)
[(2, 5, 1)]
sage: q0, = M.primes_above(Q0)
sage: q0.residue_class_degree()
5
sage: q0.relative_ramification_index()
2
sage: M.decomposition_type(Q1)
[(2, 5, 1)]

Python

>>> from sage.all import *
>>> Q0, Q1 = K.primes_above(Integer(2))
>>> M.decomposition_type(Q0)
[(2, 5, 1)]
>>> q0, = M.primes_above(Q0)
>>> q0.residue_class_degree()
5
>>> q0.relative_ramification_index()
2
>>> M.decomposition_type(Q1)
[(2, 5, 1)]

Check that Issue #34514 is fixed:

Sage

sage: K.<a> = NumberField(x^4 + 18*x^2 - 1)
sage: R.<y> = K[]
sage: L.<b> = K.extension(y^2 + 9*a^3 - 2*a^2 + 162*a - 38)
sage: [L.decomposition_type(i) for i in K.primes_above(3)]
[[(1, 1, 2)], [(1, 1, 2)], [(1, 2, 1)]]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(4) + Integer(18)*x**Integer(2) - Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> R = K['y']; (y,) = R._first_ngens(1)
>>> L = K.extension(y**Integer(2) + Integer(9)*a**Integer(3) - Integer(2)*a**Integer(2) + Integer(162)*a - Integer(38), names=('b',)); (b,) = L._first_ngens(1)
>>> [L.decomposition_type(i) for i in K.primes_above(Integer(3))]
[[(1, 1, 2)], [(1, 1, 2)], [(1, 2, 1)]]

defining_polynomial()[source]¶

Return the defining polynomial of this number field.

This is exactly the same as polynomial().

EXAMPLES:

Sage

sage: k5.<z> = CyclotomicField(5)
sage: k5.defining_polynomial()
x^4 + x^3 + x^2 + x + 1
sage: y = polygen(QQ, 'y')
sage: k.<a> = NumberField(y^9 - 3*y + 5); k
Number Field in a with defining polynomial y^9 - 3*y + 5
sage: k.defining_polynomial()
y^9 - 3*y + 5

Python

>>> from sage.all import *
>>> k5 = CyclotomicField(Integer(5), names=('z',)); (z,) = k5._first_ngens(1)
>>> k5.defining_polynomial()
x^4 + x^3 + x^2 + x + 1
>>> y = polygen(QQ, 'y')
>>> k = NumberField(y**Integer(9) - Integer(3)*y + Integer(5), names=('a',)); (a,) = k._first_ngens(1); k
Number Field in a with defining polynomial y^9 - 3*y + 5
>>> k.defining_polynomial()
y^9 - 3*y + 5

degree()[source]¶

Return the degree of this number field.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').degree()
3
sage: NumberField(x + 1, 'a').degree()
1
sage: NumberField(x^997 + 17*x + 3, 'a', check=False).degree()
997

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(3) + x**Integer(2) + Integer(997)*x + Integer(1), 'a').degree()
3
>>> NumberField(x + Integer(1), 'a').degree()
1
>>> NumberField(x**Integer(997) + Integer(17)*x + Integer(3), 'a', check=False).degree()
997

different()[source]¶

Compute the different fractional ideal of this number field.

The codifferent is the fractional ideal of all \(x\) in \(K\) such that the trace of \(xy\) is an integer for all \(y \in O_K\).

The different is the integral ideal which is the inverse of the codifferent.

See Wikipedia article Different_ideal

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: d = k.different()
sage: d
Fractional ideal (-a)
sage: d.norm()
23
sage: k.disc()
-23

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = k._first_ngens(1)
>>> d = k.different()
>>> d
Fractional ideal (-a)
>>> d.norm()
23
>>> k.disc()
-23

The different is cached:

Sage

sage: d is k.different()
True

Python

>>> from sage.all import *
>>> d is k.different()
True

Another example:

Sage

sage: k.<b> = NumberField(x^2 - 123)
sage: d = k.different(); d
Fractional ideal (2*b)
sage: d.norm()
492
sage: k.disc()
492

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(2) - Integer(123), names=('b',)); (b,) = k._first_ngens(1)
>>> d = k.different(); d
Fractional ideal (2*b)
>>> d.norm()
492
>>> k.disc()
492

dirichlet_group()[source]¶

Given a abelian field \(K\), compute and return the set of all Dirichlet characters corresponding to the characters of the Galois group of \(K/\QQ\).

The output is random if the field is not abelian.

OUTPUT: list of Dirichlet characters

EXAMPLES:

Sage

sage: # needs sage.groups sage.modular
sage: x = polygen(QQ, 'x')
sage: K.<t> = NumberField(x^3 + x^2 - 36*x - 4)
sage: K.conductor()
109
sage: K.dirichlet_group()  # optional - gap_package_polycyclic
[Dirichlet character modulo 109 of conductor 1 mapping 6 |--> 1,
 Dirichlet character modulo 109 of conductor 109 mapping 6 |--> zeta3,
 Dirichlet character modulo 109 of conductor 109 mapping 6 |--> -zeta3 - 1]

sage: # needs sage.modular
sage: K = CyclotomicField(44)
sage: L = K.subfields(5)[0][0]
sage: X = L.dirichlet_group(); X  # optional - gap_package_polycyclic
[Dirichlet character modulo 11 of conductor 1 mapping 2 |--> 1,
 Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta5,
 Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta5^2,
 Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta5^3,
 Dirichlet character modulo 11 of conductor 11
   mapping 2 |--> -zeta5^3 - zeta5^2 - zeta5 - 1]
sage: X[4]^2  # optional - gap_package_polycyclic
Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta5^3
sage: X[4]^2 in X  # optional - gap_package_polycyclic
True

Python

>>> from sage.all import *
>>> # needs sage.groups sage.modular
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(36)*x - Integer(4), names=('t',)); (t,) = K._first_ngens(1)
>>> K.conductor()
109
>>> K.dirichlet_group()  # optional - gap_package_polycyclic
[Dirichlet character modulo 109 of conductor 1 mapping 6 |--> 1,
 Dirichlet character modulo 109 of conductor 109 mapping 6 |--> zeta3,
 Dirichlet character modulo 109 of conductor 109 mapping 6 |--> -zeta3 - 1]

>>> # needs sage.modular
>>> K = CyclotomicField(Integer(44))
>>> L = K.subfields(Integer(5))[Integer(0)][Integer(0)]
>>> X = L.dirichlet_group(); X  # optional - gap_package_polycyclic
[Dirichlet character modulo 11 of conductor 1 mapping 2 |--> 1,
 Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta5,
 Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta5^2,
 Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta5^3,
 Dirichlet character modulo 11 of conductor 11
   mapping 2 |--> -zeta5^3 - zeta5^2 - zeta5 - 1]
>>> X[Integer(4)]**Integer(2)  # optional - gap_package_polycyclic
Dirichlet character modulo 11 of conductor 11 mapping 2 |--> zeta5^3
>>> X[Integer(4)]**Integer(2) in X  # optional - gap_package_polycyclic
True

disc(v=None)[source]¶

Shortcut for discriminant().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<b> = NumberField(x^2 - 123)
sage: k.disc()
492

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(2) - Integer(123), names=('b',)); (b,) = k._first_ngens(1)
>>> k.disc()
492

discriminant(v=None)[source]¶

Return the discriminant of the ring of integers of the number field, or if v is specified, the determinant of the trace pairing on the elements of the list v.

INPUT:

v – (optional) list of elements of this number field

OUTPUT: integer if v is omitted, and Rational otherwise

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<t> = NumberField(x^3 + x^2 - 2*x + 8)
sage: K.disc()
-503
sage: K.disc([1, t, t^2])
-2012
sage: K.disc([1/7, (1/5)*t, (1/3)*t^2])
-2012/11025
sage: (5*7*3)^2
11025
sage: NumberField(x^2 - 1/2, 'a').discriminant()
8

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x + Integer(8), names=('t',)); (t,) = K._first_ngens(1)
>>> K.disc()
-503
>>> K.disc([Integer(1), t, t**Integer(2)])
-2012
>>> K.disc([Integer(1)/Integer(7), (Integer(1)/Integer(5))*t, (Integer(1)/Integer(3))*t**Integer(2)])
-2012/11025
>>> (Integer(5)*Integer(7)*Integer(3))**Integer(2)
11025
>>> NumberField(x**Integer(2) - Integer(1)/Integer(2), 'a').discriminant()
8

elements_of_norm(n, proof=None)[source]¶

Return a list of elements of norm \(n\).

INPUT:

n – integer
proof – boolean (default: True, unless you called proof.number_field() and set it otherwise)

OUTPUT:

A complete system of integral elements of norm \(n\), modulo units of positive norm.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 + 1)
sage: K.elements_of_norm(3)
[]
sage: K.elements_of_norm(50)
[-a - 7, 5*a - 5, 7*a + 1]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.elements_of_norm(Integer(3))
[]
>>> K.elements_of_norm(Integer(50))
[-a - 7, 5*a - 5, 7*a + 1]

extension(poly, name=None, names=None, latex_name=None, latex_names=None, *args, **kwds)[source]¶

Return the relative extension of this field by a given polynomial.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: R.<t> = K[]
sage: L.<b> = K.extension(t^2 + a); L
Number Field in b with defining polynomial t^2 + a over its base field

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> R = K['t']; (t,) = R._first_ngens(1)
>>> L = K.extension(t**Integer(2) + a, names=('b',)); (b,) = L._first_ngens(1); L
Number Field in b with defining polynomial t^2 + a over its base field

We create another extension:

Sage

sage: k.<a> = NumberField(x^2 + 1); k
Number Field in a with defining polynomial x^2 + 1
sage: y = polygen(QQ,'y')
sage: m.<b> = k.extension(y^2 + 2); m
Number Field in b with defining polynomial y^2 + 2 over its base field

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = k._first_ngens(1); k
Number Field in a with defining polynomial x^2 + 1
>>> y = polygen(QQ,'y')
>>> m = k.extension(y**Integer(2) + Integer(2), names=('b',)); (b,) = m._first_ngens(1); m
Number Field in b with defining polynomial y^2 + 2 over its base field

Note that \(b\) is a root of \(y^2 + 2\):

Sage

sage: b.minpoly()
x^2 + 2
sage: b.minpoly('z')
z^2 + 2

Python

>>> from sage.all import *
>>> b.minpoly()
x^2 + 2
>>> b.minpoly('z')
z^2 + 2

A relative extension of a relative extension:

Sage

sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1])
sage: R.<z> = k[]
sage: L.<b> = NumberField(z^3 + 3 + a); L
Number Field in b with defining polynomial z^3 + a0 + 3 over its base field

Python

>>> from sage.all import *
>>> k = NumberField([x**Integer(2) + Integer(1), x**Integer(3) + x + Integer(1)], names=('a',)); (a,) = k._first_ngens(1)
>>> R = k['z']; (z,) = R._first_ngens(1)
>>> L = NumberField(z**Integer(3) + Integer(3) + a, names=('b',)); (b,) = L._first_ngens(1); L
Number Field in b with defining polynomial z^3 + a0 + 3 over its base field

Extension fields with given defining data are unique (Issue #20791):

Sage

sage: K.<a> = NumberField(x^2 + 1)
sage: K.extension(x^2 - 2, 'b') is K.extension(x^2 - 2, 'b')
True

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.extension(x**Integer(2) - Integer(2), 'b') is K.extension(x**Integer(2) - Integer(2), 'b')
True

factor(n)[source]¶

Ideal factorization of the principal ideal generated by \(n\).

EXAMPLES:

Here we show how to factor Gaussian integers (up to units). First we form a number field defined by \(x^2 + 1\):

Sage

sage: x = polygen(QQ, 'x')
sage: K.<I> = NumberField(x^2 + 1); K
Number Field in I with defining polynomial x^2 + 1

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('I',)); (I,) = K._first_ngens(1); K
Number Field in I with defining polynomial x^2 + 1

Here are the factors:

Sage

sage: fi, fj = K.factor(17); fi,fj
((Fractional ideal (I + 4), 1), (Fractional ideal (I - 4), 1))

Python

>>> from sage.all import *
>>> fi, fj = K.factor(Integer(17)); fi,fj
((Fractional ideal (I + 4), 1), (Fractional ideal (I - 4), 1))

Now we extract the reduced form of the generators:

Sage

sage: zi = fi[0].gens_reduced()[0]; zi
I + 4
sage: zj = fj[0].gens_reduced()[0]; zj
I - 4

Python

>>> from sage.all import *
>>> zi = fi[Integer(0)].gens_reduced()[Integer(0)]; zi
I + 4
>>> zj = fj[Integer(0)].gens_reduced()[Integer(0)]; zj
I - 4

We recover the integer that was factored in \(\ZZ[i]\) (up to a unit):

Sage

sage: zi*zj
-17

Python

>>> from sage.all import *
>>> zi*zj
-17

One can also factor elements or ideals of the number field:

Sage

sage: K.<a> = NumberField(x^2 + 1)
sage: K.factor(1/3)
(Fractional ideal (3))^-1
sage: K.factor(1+a)
Fractional ideal (a + 1)
sage: K.factor(1+a/5)
(Fractional ideal (a + 1)) * (Fractional ideal (-a - 2))^-1
  * (Fractional ideal (2*a + 1))^-1 * (Fractional ideal (-2*a + 3))

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.factor(Integer(1)/Integer(3))
(Fractional ideal (3))^-1
>>> K.factor(Integer(1)+a)
Fractional ideal (a + 1)
>>> K.factor(Integer(1)+a/Integer(5))
(Fractional ideal (a + 1)) * (Fractional ideal (-a - 2))^-1
  * (Fractional ideal (2*a + 1))^-1 * (Fractional ideal (-2*a + 3))

An example over a relative number field:

Sage

sage: pari('setrand(2)')
sage: L.<b> = K.extension(x^2 - 7)
sage: f = L.factor(a + 1)
sage: f                               # representation varies, not tested
(Fractional ideal (1/2*a*b - a + 1/2)) * (Fractional ideal (-1/2*a*b - a + 1/2))
sage: f.value() == a+1
True

Python

>>> from sage.all import *
>>> pari('setrand(2)')
>>> L = K.extension(x**Integer(2) - Integer(7), names=('b',)); (b,) = L._first_ngens(1)
>>> f = L.factor(a + Integer(1))
>>> f                               # representation varies, not tested
(Fractional ideal (1/2*a*b - a + 1/2)) * (Fractional ideal (-1/2*a*b - a + 1/2))
>>> f.value() == a+Integer(1)
True

It doesn’t make sense to factor the ideal \((0)\), so this raises an error:

Sage

sage: L.factor(0)
Traceback (most recent call last):
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'factor'...

Python

>>> from sage.all import *
>>> L.factor(Integer(0))
Traceback (most recent call last):
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'factor'...

AUTHORS:

Alex Clemesha (2006-05-20), Francis Clarke (2009-04-21): examples

fractional_ideal(*gens, **kwds)[source]¶

Return the ideal in \(\mathcal{O}_K\) generated by gens.

This overrides the sage.rings.ring.Field method to use the sage.rings.ring.Ring one instead, since we are not concerned with ideals in a field but in its ring of integers.

INPUT:

gens – list of generators, or a number field ideal

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: K.fractional_ideal([1/a])
Fractional ideal (1/2*a^2)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.fractional_ideal([Integer(1)/a])
Fractional ideal (1/2*a^2)

One can also input a number field ideal itself, or, more usefully, for a tower of number fields an ideal in one of the fields lower down the tower.

Sage

sage: K.fractional_ideal(K.ideal(a))
Fractional ideal (a)
sage: L.<b> = K.extension(x^2 - 3, x^2 + 1)
sage: M.<c> = L.extension(x^2 + 1)
sage: L.ideal(K.ideal(2, a))
Fractional ideal (a)
sage: M.ideal(K.ideal(2, a)) == M.ideal(a*(b - c)/2)
True

Python

>>> from sage.all import *
>>> K.fractional_ideal(K.ideal(a))
Fractional ideal (a)
>>> L = K.extension(x**Integer(2) - Integer(3), x**Integer(2) + Integer(1), names=('b',)); (b,) = L._first_ngens(1)
>>> M = L.extension(x**Integer(2) + Integer(1), names=('c',)); (c,) = M._first_ngens(1)
>>> L.ideal(K.ideal(Integer(2), a))
Fractional ideal (a)
>>> M.ideal(K.ideal(Integer(2), a)) == M.ideal(a*(b - c)/Integer(2))
True

The zero ideal is not a fractional ideal!

Sage

sage: K.fractional_ideal(0)
Traceback (most recent call last):
...
ValueError: gens must have a nonzero element (zero ideal is not a fractional ideal)

Python

>>> from sage.all import *
>>> K.fractional_ideal(Integer(0))
Traceback (most recent call last):
...
ValueError: gens must have a nonzero element (zero ideal is not a fractional ideal)

galois_group(type=None, algorithm='pari', names=None, gc_numbering=None)[source]¶

Return the Galois group of the Galois closure of this number field.

INPUT:

type – deprecated; the different versions of Galois groups have been merged in Issue #28782
algorithm – 'pari', 'gap', 'kash', or 'magma' (default: 'pari'); for degrees between 12 and 15 default is 'gap', and when the degree is >= 16 it is 'kash')
names – string giving a name for the generator of the Galois closure of self, when this field is not Galois
gc_numbering – if True, permutations will be written in terms of the action on the roots of a defining polynomial for the Galois closure, rather than the defining polynomial for the original number field. This is significantly faster; but not the standard way of presenting Galois groups. The default currently depends on the algorithm (True for 'pari', False for 'magma') and may change in the future.

The resulting group will only compute with automorphisms when necessary, so certain functions (such as sage.rings.number_field.galois_group.GaloisGroup_v2.order()) will still be fast. For more (important!) documentation, see the documentation for Galois groups of polynomials over \(\QQ\), e.g., by typing K.polynomial().galois_group?, where \(K\) is a number field.

EXAMPLES:

Sage

sage: # needs sage.groups
sage: x = polygen(QQ, 'x')
sage: k.<b> = NumberField(x^2 - 14)  # a Galois extension
sage: G = k.galois_group(); G
Galois group 2T1 (S2) with order 2 of x^2 - 14
sage: G.gen(0)
(1,2)
sage: G.gen(0)(b)
-b
sage: G.artin_symbol(k.primes_above(3)[0])
(1,2)

sage: # needs sage.groups
sage: k.<b> = NumberField(x^3 - x + 1)  # not Galois
sage: G = k.galois_group(names='c'); G
Galois group 3T2 (S3) with order 6 of x^3 - x + 1
sage: G.gen(0)
(1,2,3)(4,5,6)

sage: NumberField(x^3 + 2*x + 1, 'a').galois_group(algorithm='magma')   # optional - magma, needs sage.groups
Galois group Transitive group number 2 of degree 3
 of the Number Field in a with defining polynomial x^3 + 2*x + 1

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(2) - Integer(14), names=('b',)); (b,) = k._first_ngens(1)# a Galois extension
>>> G = k.galois_group(); G
Galois group 2T1 (S2) with order 2 of x^2 - 14
>>> G.gen(Integer(0))
(1,2)
>>> G.gen(Integer(0))(b)
-b
>>> G.artin_symbol(k.primes_above(Integer(3))[Integer(0)])
(1,2)

>>> # needs sage.groups
>>> k = NumberField(x**Integer(3) - x + Integer(1), names=('b',)); (b,) = k._first_ngens(1)# not Galois
>>> G = k.galois_group(names='c'); G
Galois group 3T2 (S3) with order 6 of x^3 - x + 1
>>> G.gen(Integer(0))
(1,2,3)(4,5,6)

>>> NumberField(x**Integer(3) + Integer(2)*x + Integer(1), 'a').galois_group(algorithm='magma')   # optional - magma, needs sage.groups
Galois group Transitive group number 2 of degree 3
 of the Number Field in a with defining polynomial x^3 + 2*x + 1

EXPLICIT GALOIS GROUP: We compute the Galois group as an explicit group of automorphisms of the Galois closure of a field.

Sage

sage: # needs sage.groups
sage: K.<a> = NumberField(x^3 - 2)
sage: L.<b1> = K.galois_closure(); L
Number Field in b1 with defining polynomial x^6 + 108
sage: G = End(L); G
Automorphism group of Number Field in b1 with defining polynomial x^6 + 108
sage: G.list()
[
Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 108
  Defn: b1 |--> b1,
...
Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 108
  Defn: b1 |--> -1/12*b1^4 - 1/2*b1
]
sage: G[2](b1)
1/12*b1^4 + 1/2*b1

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.galois_closure(names=('b1',)); (b1,) = L._first_ngens(1); L
Number Field in b1 with defining polynomial x^6 + 108
>>> G = End(L); G
Automorphism group of Number Field in b1 with defining polynomial x^6 + 108
>>> G.list()
[
Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 108
  Defn: b1 |--> b1,
...
Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 108
  Defn: b1 |--> -1/12*b1^4 - 1/2*b1
]
>>> G[Integer(2)](b1)
1/12*b1^4 + 1/2*b1

Many examples for higher degrees may be found in the online databases http://galoisdb.math.upb.de/ by Jürgen Klüners and Gunter Malle and https://www.lmfdb.org/NumberField/ by the LMFDB collaboration, although these might need a lot of computing time.

If \(L/K\) is a relative number field, this method will currently return \(Gal(L/\QQ)\). This behavior will change in the future, so it’s better to explicitly call absolute_field() if that is the desired behavior:

Sage

sage: # needs sage.groups
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2 + 1)
sage: R.<t> = PolynomialRing(K)
sage: L = K.extension(t^5 - t + a, 'b')
sage: L.galois_group()
...DeprecationWarning: Use .absolute_field().galois_group()
if you want the Galois group of the absolute field
See https://github.com/sagemath/sage/issues/28782 for details.
Galois group 10T22 (S(5)[x]2) with order 240 of t^5 - t + a

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> x = polygen(QQ)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> R = PolynomialRing(K, names=('t',)); (t,) = R._first_ngens(1)
>>> L = K.extension(t**Integer(5) - t + a, 'b')
>>> L.galois_group()
...DeprecationWarning: Use .absolute_field().galois_group()
if you want the Galois group of the absolute field
See https://github.com/sagemath/sage/issues/28782 for details.
Galois group 10T22 (S(5)[x]2) with order 240 of t^5 - t + a

gen(n=0)[source]¶

Return the generator for this number field.

INPUT:

n – must be 0 (the default), or an exception is raised

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<theta> = NumberField(x^14 + 2); k
Number Field in theta with defining polynomial x^14 + 2
sage: k.gen()
theta
sage: k.gen(1)
Traceback (most recent call last):
...
IndexError: Only one generator.

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(14) + Integer(2), names=('theta',)); (theta,) = k._first_ngens(1); k
Number Field in theta with defining polynomial x^14 + 2
>>> k.gen()
theta
>>> k.gen(Integer(1))
Traceback (most recent call last):
...
IndexError: Only one generator.

gen_embedding()[source]¶

If an embedding has been specified, return the image of the generator under that embedding. Otherwise return None.

EXAMPLES:

Sage

sage: QuadraticField(-7, 'a').gen_embedding()
2.645751311064591?*I
sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + 7, 'a').gen_embedding()  # None

Python

>>> from sage.all import *
>>> QuadraticField(-Integer(7), 'a').gen_embedding()
2.645751311064591?*I
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + Integer(7), 'a').gen_embedding()  # None

ideal(*gens, **kwds)[source]¶

Return a fractional ideal of the field, except for the zero ideal, which is not a fractional ideal.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(2)
Fractional ideal (2)
sage: K.ideal(2 + i)
Fractional ideal (i + 2)
sage: K.ideal(0)
Ideal (0) of Number Field in i with defining polynomial x^2 + 1

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.ideal(Integer(2))
Fractional ideal (2)
>>> K.ideal(Integer(2) + i)
Fractional ideal (i + 2)
>>> K.ideal(Integer(0))
Ideal (0) of Number Field in i with defining polynomial x^2 + 1

idealchinese(ideals, residues)[source]¶

Return a solution of the Chinese Remainder Theorem problem for ideals in a number field.

This is a wrapper around the pari function pari:idealchinese.

INPUT:

ideals – list of ideals of the number field
residues – list of elements of the number field

OUTPUT:

Return an element \(b\) of the number field such that \(b \equiv x_i \bmod I_i\) for all residues \(x_i\) and respective ideals \(I_i\).

See also

crt()

EXAMPLES:

This is the example from the pari page on idealchinese:

Sage

sage: # needs sage.symbolic
sage: K.<sqrt2> = NumberField(sqrt(2).minpoly())
sage: ideals = [K.ideal(4), K.ideal(3)]
sage: residues = [sqrt2, 1]
sage: r = K.idealchinese(ideals, residues); r
-3*sqrt2 + 4
sage: all((r - a) in I for I, a in zip(ideals, residues))
True

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> K = NumberField(sqrt(Integer(2)).minpoly(), names=('sqrt2',)); (sqrt2,) = K._first_ngens(1)
>>> ideals = [K.ideal(Integer(4)), K.ideal(Integer(3))]
>>> residues = [sqrt2, Integer(1)]
>>> r = K.idealchinese(ideals, residues); r
-3*sqrt2 + 4
>>> all((r - a) in I for I, a in zip(ideals, residues))
True

The result may be non-integral if the results are non-integral:

Sage

sage: # needs sage.symbolic
sage: K.<sqrt2> = NumberField(sqrt(2).minpoly())
sage: ideals = [K.ideal(4), K.ideal(21)]
sage: residues = [1/sqrt2, 1]
sage: r = K.idealchinese(ideals, residues); r
-63/2*sqrt2 - 20
sage: all(
....:     (r - a).valuation(P) >= k
....:     for I, a in zip(ideals, residues)
....:     for P, k in I.factor()
....: )
True

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> K = NumberField(sqrt(Integer(2)).minpoly(), names=('sqrt2',)); (sqrt2,) = K._first_ngens(1)
>>> ideals = [K.ideal(Integer(4)), K.ideal(Integer(21))]
>>> residues = [Integer(1)/sqrt2, Integer(1)]
>>> r = K.idealchinese(ideals, residues); r
-63/2*sqrt2 - 20
>>> all(
...     (r - a).valuation(P) >= k
...     for I, a in zip(ideals, residues)
...     for P, k in I.factor()
... )
True

ideals_of_bdd_norm(bound)[source]¶

Return all integral ideals of bounded norm.

INPUT:

bound – positive integer

OUTPUT: a dict of all integral ideals \(I\) such that Norm(\(I\)) \(\leq\) bound, keyed by norm.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 + 23)
sage: d = K.ideals_of_bdd_norm(10)
sage: for n in d:
....:     print(n)
....:     for I in sorted(d[n]):
....:         print(I)
1
Fractional ideal (1)
2
Fractional ideal (2, 1/2*a - 1/2)
Fractional ideal (2, 1/2*a + 1/2)
3
Fractional ideal (3, 1/2*a - 1/2)
Fractional ideal (3, 1/2*a + 1/2)
4
Fractional ideal (2)
Fractional ideal (4, 1/2*a + 3/2)
Fractional ideal (4, 1/2*a + 5/2)
5
6
Fractional ideal (1/2*a - 1/2)
Fractional ideal (1/2*a + 1/2)
Fractional ideal (6, 1/2*a + 5/2)
Fractional ideal (6, 1/2*a + 7/2)
7
8
Fractional ideal (4, a - 1)
Fractional ideal (4, a + 1)
Fractional ideal (1/2*a + 3/2)
Fractional ideal (1/2*a - 3/2)
9
Fractional ideal (3)
Fractional ideal (9, 1/2*a + 7/2)
Fractional ideal (9, 1/2*a + 11/2)
10

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> d = K.ideals_of_bdd_norm(Integer(10))
>>> for n in d:
...     print(n)
...     for I in sorted(d[n]):
...         print(I)
1
Fractional ideal (1)
2
Fractional ideal (2, 1/2*a - 1/2)
Fractional ideal (2, 1/2*a + 1/2)
3
Fractional ideal (3, 1/2*a - 1/2)
Fractional ideal (3, 1/2*a + 1/2)
4
Fractional ideal (2)
Fractional ideal (4, 1/2*a + 3/2)
Fractional ideal (4, 1/2*a + 5/2)
5
6
Fractional ideal (1/2*a - 1/2)
Fractional ideal (1/2*a + 1/2)
Fractional ideal (6, 1/2*a + 5/2)
Fractional ideal (6, 1/2*a + 7/2)
7
8
Fractional ideal (4, a - 1)
Fractional ideal (4, a + 1)
Fractional ideal (1/2*a + 3/2)
Fractional ideal (1/2*a - 3/2)
9
Fractional ideal (3)
Fractional ideal (9, 1/2*a + 7/2)
Fractional ideal (9, 1/2*a + 11/2)
10

integral_basis(v=None)[source]¶

Return a list containing a ZZ-basis for the full ring of integers of this number field.

INPUT:

v – None, a prime, or a list of primes; see the documentation for maximal_order()

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^5 + 10*x + 1)
sage: K.integral_basis()
[1, a, a^2, a^3, a^4]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(5) + Integer(10)*x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.integral_basis()
[1, a, a^2, a^3, a^4]

Next we compute the ring of integers of a cubic field in which 2 is an “essential discriminant divisor”, so the ring of integers is not generated by a single element.

Sage

sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: K.integral_basis()
[1, 1/2*a^2 + 1/2*a, a^2]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x + Integer(8), names=('a',)); (a,) = K._first_ngens(1)
>>> K.integral_basis()
[1, 1/2*a^2 + 1/2*a, a^2]

ALGORITHM: Uses the PARI library (via pari:_pari_integral_basis).

is_CM()[source]¶

Return True if self is a CM field (i.e., a totally imaginary quadratic extension of a totally real field).

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: Q.<a> = NumberField(x - 1)
sage: Q.is_CM()
False
sage: K.<i> = NumberField(x^2 + 1)
sage: K.is_CM()
True
sage: L.<zeta20> = CyclotomicField(20)
sage: L.is_CM()
True
sage: K.<omega> = QuadraticField(-3)
sage: K.is_CM()
True
sage: L.<sqrt5> = QuadraticField(5)
sage: L.is_CM()
False
sage: F.<a> = NumberField(x^3 - 2)
sage: F.is_CM()
False
sage: F.<a> = NumberField(x^4 - x^3 - 3*x^2 + x + 1)
sage: F.is_CM()
False

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> Q = NumberField(x - Integer(1), names=('a',)); (a,) = Q._first_ngens(1)
>>> Q.is_CM()
False
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.is_CM()
True
>>> L = CyclotomicField(Integer(20), names=('zeta20',)); (zeta20,) = L._first_ngens(1)
>>> L.is_CM()
True
>>> K = QuadraticField(-Integer(3), names=('omega',)); (omega,) = K._first_ngens(1)
>>> K.is_CM()
True
>>> L = QuadraticField(Integer(5), names=('sqrt5',)); (sqrt5,) = L._first_ngens(1)
>>> L.is_CM()
False
>>> F = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> F.is_CM()
False
>>> F = NumberField(x**Integer(4) - x**Integer(3) - Integer(3)*x**Integer(2) + x + Integer(1), names=('a',)); (a,) = F._first_ngens(1)
>>> F.is_CM()
False

The following are non-CM totally imaginary fields.

Sage

sage: F.<a> = NumberField(x^4 + x^3 - x^2 - x + 1)
sage: F.is_totally_imaginary()
True
sage: F.is_CM()
False
sage: F2.<a> = NumberField(x^12 - 5*x^11 + 8*x^10 - 5*x^9 - x^8 + 9*x^7
....:                       + 7*x^6 - 3*x^5 + 5*x^4 + 7*x^3 - 4*x^2 - 7*x + 7)
sage: F2.is_totally_imaginary()
True
sage: F2.is_CM()
False

Python

>>> from sage.all import *
>>> F = NumberField(x**Integer(4) + x**Integer(3) - x**Integer(2) - x + Integer(1), names=('a',)); (a,) = F._first_ngens(1)
>>> F.is_totally_imaginary()
True
>>> F.is_CM()
False
>>> F2 = NumberField(x**Integer(12) - Integer(5)*x**Integer(11) + Integer(8)*x**Integer(10) - Integer(5)*x**Integer(9) - x**Integer(8) + Integer(9)*x**Integer(7)
...                       + Integer(7)*x**Integer(6) - Integer(3)*x**Integer(5) + Integer(5)*x**Integer(4) + Integer(7)*x**Integer(3) - Integer(4)*x**Integer(2) - Integer(7)*x + Integer(7), names=('a',)); (a,) = F2._first_ngens(1)
>>> F2.is_totally_imaginary()
True
>>> F2.is_CM()
False

The following is a non-cyclotomic CM field.

Sage

sage: M.<a> = NumberField(x^4 - x^3 - x^2 - 2*x + 4)
sage: M.is_CM()
True

Python

>>> from sage.all import *
>>> M = NumberField(x**Integer(4) - x**Integer(3) - x**Integer(2) - Integer(2)*x + Integer(4), names=('a',)); (a,) = M._first_ngens(1)
>>> M.is_CM()
True

Now, we construct a totally imaginary quadratic extension of a totally real field (which is not cyclotomic).

Sage

sage: E_0.<a> = NumberField(x^7 - 4*x^6 - 4*x^5 + 10*x^4 + 4*x^3
....:                       - 6*x^2 - x + 1)
sage: E_0.is_totally_real()
True
sage: E.<b> = E_0.extension(x^2 + 1)
sage: E.is_CM()
True

Python

>>> from sage.all import *
>>> E_0 = NumberField(x**Integer(7) - Integer(4)*x**Integer(6) - Integer(4)*x**Integer(5) + Integer(10)*x**Integer(4) + Integer(4)*x**Integer(3)
...                       - Integer(6)*x**Integer(2) - x + Integer(1), names=('a',)); (a,) = E_0._first_ngens(1)
>>> E_0.is_totally_real()
True
>>> E = E_0.extension(x**Integer(2) + Integer(1), names=('b',)); (b,) = E._first_ngens(1)
>>> E.is_CM()
True

Finally, a CM field that is given as an extension that is not CM.

Sage

sage: E_0.<a> = NumberField(x^2 - 4*x + 16)
sage: y = polygen(E_0)
sage: E.<z> = E_0.extension(y^2 - E_0.gen() / 2)
sage: E.is_CM()
True
sage: E.is_CM_extension()
False

Python

>>> from sage.all import *
>>> E_0 = NumberField(x**Integer(2) - Integer(4)*x + Integer(16), names=('a',)); (a,) = E_0._first_ngens(1)
>>> y = polygen(E_0)
>>> E = E_0.extension(y**Integer(2) - E_0.gen() / Integer(2), names=('z',)); (z,) = E._first_ngens(1)
>>> E.is_CM()
True
>>> E.is_CM_extension()
False

is_abelian()[source]¶

Return True if this number field is an abelian Galois extension of \(\QQ\).

EXAMPLES:

Sage

sage: # needs sage.groups
sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + 1, 'i').is_abelian()
True
sage: NumberField(x^3 + 2, 'a').is_abelian()
False
sage: NumberField(x^3 + x^2 - 2*x - 1, 'a').is_abelian()
True
sage: NumberField(x^6 + 40*x^3 + 1372, 'a').is_abelian()
False
sage: NumberField(x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1, 'a').is_abelian()
True

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + Integer(1), 'i').is_abelian()
True
>>> NumberField(x**Integer(3) + Integer(2), 'a').is_abelian()
False
>>> NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x - Integer(1), 'a').is_abelian()
True
>>> NumberField(x**Integer(6) + Integer(40)*x**Integer(3) + Integer(1372), 'a').is_abelian()
False
>>> NumberField(x**Integer(6) + x**Integer(5) - Integer(5)*x**Integer(4) - Integer(4)*x**Integer(3) + Integer(6)*x**Integer(2) + Integer(3)*x - Integer(1), 'a').is_abelian()
True

is_absolute()[source]¶

Return True if self is an absolute field.

This function will be implemented in the derived classes.

EXAMPLES:

Sage

sage: K = CyclotomicField(5)
sage: K.is_absolute()
True

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(5))
>>> K.is_absolute()
True

is_field(proof=True)[source]¶

Return True since a number field is a field.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^5 + x + 3, 'c').is_field()
True

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(5) + x + Integer(3), 'c').is_field()
True

is_galois()[source]¶

Return True if this number field is a Galois extension of \(\QQ\).

EXAMPLES:

Sage

sage: # needs sage.groups
sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + 1, 'i').is_galois()
True
sage: NumberField(x^3 + 2, 'a').is_galois()
False
sage: K = NumberField(x^15 + x^14 - 14*x^13 - 13*x^12 + 78*x^11 + 66*x^10
....:                  - 220*x^9 - 165*x^8 + 330*x^7 + 210*x^6 - 252*x^5
....:                  - 126*x^4 + 84*x^3 + 28*x^2 - 8*x - 1, 'a')
sage: K.is_galois()
True
sage: K = NumberField(x^15 + x^14 - 14*x^13 - 13*x^12 + 78*x^11 + 66*x^10
....:                  - 220*x^9 - 165*x^8 + 330*x^7 + 210*x^6 - 252*x^5
....:                  - 126*x^4 + 84*x^3 + 28*x^2 - 8*x - 10, 'a')
sage: K.is_galois()
False

Python

>>> from sage.all import *
>>> # needs sage.groups
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + Integer(1), 'i').is_galois()
True
>>> NumberField(x**Integer(3) + Integer(2), 'a').is_galois()
False
>>> K = NumberField(x**Integer(15) + x**Integer(14) - Integer(14)*x**Integer(13) - Integer(13)*x**Integer(12) + Integer(78)*x**Integer(11) + Integer(66)*x**Integer(10)
...                  - Integer(220)*x**Integer(9) - Integer(165)*x**Integer(8) + Integer(330)*x**Integer(7) + Integer(210)*x**Integer(6) - Integer(252)*x**Integer(5)
...                  - Integer(126)*x**Integer(4) + Integer(84)*x**Integer(3) + Integer(28)*x**Integer(2) - Integer(8)*x - Integer(1), 'a')
>>> K.is_galois()
True
>>> K = NumberField(x**Integer(15) + x**Integer(14) - Integer(14)*x**Integer(13) - Integer(13)*x**Integer(12) + Integer(78)*x**Integer(11) + Integer(66)*x**Integer(10)
...                  - Integer(220)*x**Integer(9) - Integer(165)*x**Integer(8) + Integer(330)*x**Integer(7) + Integer(210)*x**Integer(6) - Integer(252)*x**Integer(5)
...                  - Integer(126)*x**Integer(4) + Integer(84)*x**Integer(3) + Integer(28)*x**Integer(2) - Integer(8)*x - Integer(10), 'a')
>>> K.is_galois()
False

is_isomorphic(other, isomorphism_maps=False)[source]¶

Return True if self is isomorphic as a number field to other.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 1)
sage: m.<b> = NumberField(x^2 + 4)
sage: k.is_isomorphic(m)
True
sage: m.<b> = NumberField(x^2 + 5)
sage: k.is_isomorphic (m)
False

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = k._first_ngens(1)
>>> m = NumberField(x**Integer(2) + Integer(4), names=('b',)); (b,) = m._first_ngens(1)
>>> k.is_isomorphic(m)
True
>>> m = NumberField(x**Integer(2) + Integer(5), names=('b',)); (b,) = m._first_ngens(1)
>>> k.is_isomorphic (m)
False

Sage

sage: k = NumberField(x^3 + 2, 'a')
sage: k.is_isomorphic(NumberField((x+1/3)^3 + 2, 'b'))
True
sage: k.is_isomorphic(NumberField(x^3 + 4, 'b'))
True
sage: k.is_isomorphic(NumberField(x^3 + 5, 'b'))
False

sage: k = NumberField(x^2 - x - 1, 'b')
sage: l = NumberField(x^2 - 7, 'a')
sage: k.is_isomorphic(l, True)
(False, [])

sage: k = NumberField(x^2 - x - 1, 'b')
sage: ky.<y> = k[]
sage: l = NumberField(y, 'a')
sage: k.is_isomorphic(l, True)
(True, [-x, x + 1])

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(3) + Integer(2), 'a')
>>> k.is_isomorphic(NumberField((x+Integer(1)/Integer(3))**Integer(3) + Integer(2), 'b'))
True
>>> k.is_isomorphic(NumberField(x**Integer(3) + Integer(4), 'b'))
True
>>> k.is_isomorphic(NumberField(x**Integer(3) + Integer(5), 'b'))
False

>>> k = NumberField(x**Integer(2) - x - Integer(1), 'b')
>>> l = NumberField(x**Integer(2) - Integer(7), 'a')
>>> k.is_isomorphic(l, True)
(False, [])

>>> k = NumberField(x**Integer(2) - x - Integer(1), 'b')
>>> ky = k['y']; (y,) = ky._first_ngens(1)
>>> l = NumberField(y, 'a')
>>> k.is_isomorphic(l, True)
(True, [-x, x + 1])

is_relative()[source]¶

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^10 - 2)
sage: K.is_absolute()
True
sage: K.is_relative()
False

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(10) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.is_absolute()
True
>>> K.is_relative()
False

is_totally_imaginary()[source]¶

Return True if self is totally imaginary, and False otherwise.

Totally imaginary means that no isomorphic embedding of self into the complex numbers has image contained in the real numbers.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + 2, 'alpha').is_totally_imaginary()
True
sage: NumberField(x^2 - 2, 'alpha').is_totally_imaginary()
False
sage: NumberField(x^4 - 2, 'alpha').is_totally_imaginary()
False

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + Integer(2), 'alpha').is_totally_imaginary()
True
>>> NumberField(x**Integer(2) - Integer(2), 'alpha').is_totally_imaginary()
False
>>> NumberField(x**Integer(4) - Integer(2), 'alpha').is_totally_imaginary()
False

is_totally_real()[source]¶

Return True if self is totally real, and False otherwise.

Totally real means that every isomorphic embedding of self into the complex numbers has image contained in the real numbers.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + 2, 'alpha').is_totally_real()
False
sage: NumberField(x^2 - 2, 'alpha').is_totally_real()
True
sage: NumberField(x^4 - 2, 'alpha').is_totally_real()
False

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + Integer(2), 'alpha').is_totally_real()
False
>>> NumberField(x**Integer(2) - Integer(2), 'alpha').is_totally_real()
True
>>> NumberField(x**Integer(4) - Integer(2), 'alpha').is_totally_real()
False

lmfdb_page()[source]¶

Open the LMFDB web page of the number field in a browser.

See https://www.lmfdb.org

EXAMPLES:

Sage

sage: E = QuadraticField(-1)
sage: E.lmfdb_page()  # optional -- webbrowser

Python

>>> from sage.all import *
>>> E = QuadraticField(-Integer(1))
>>> E.lmfdb_page()  # optional -- webbrowser

Even if the variable name is different it works:

Sage

sage: R.<y>= PolynomialRing(QQ, "y")
sage: K = NumberField(y^2 + 1 , "i")
sage: K.lmfdb_page()  # optional -- webbrowser

Python

>>> from sage.all import *
>>> R = PolynomialRing(QQ, "y", names=('y',)); (y,) = R._first_ngens(1)
>>> K = NumberField(y**Integer(2) + Integer(1) , "i")
>>> K.lmfdb_page()  # optional -- webbrowser

maximal_order(v=None, assume_maximal='non-maximal-non-unique')[source]¶

Return the maximal order, i.e., the ring of integers, associated to this number field.

INPUT:

v – None, a prime, or a list of integer primes (default: None)
- if None, return the maximal order.
- if a prime \(p\), return an order that is \(p\)-maximal.
- if a list, return an order that is maximal at each prime of these primes
assume_maximal – True, False, None, or 'non-maximal-non-unique' (default: 'non-maximal-non-unique') ignored when v is None; otherwise, controls whether we assume that the order order.is_maximal() outside of v.
- if True, the order is assumed to be maximal at all primes.
- if False, the order is assumed to be non-maximal at some prime not in v.
- if None, no assumptions are made about primes not in v.
- if 'non-maximal-non-unique' (deprecated), like False, however, the order is not a unique parent, so creating the same order later does typically not poison caches with the information that the order is not maximal.

EXAMPLES:

In this example, the maximal order cannot be generated by a single element:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^3 + x^2 - 2*x+8)
sage: o = k.maximal_order()
sage: o
Maximal Order generated by [1/2*a^2 + 1/2*a, a^2] in Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x+Integer(8), names=('a',)); (a,) = k._first_ngens(1)
>>> o = k.maximal_order()
>>> o
Maximal Order generated by [1/2*a^2 + 1/2*a, a^2] in Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8

We compute \(p\)-maximal orders for several \(p\). Note that computing a \(p\)-maximal order is much faster in general than computing the maximal order:

Sage

sage: p = next_prime(10^22)
sage: q = next_prime(10^23)
sage: K.<a> = NumberField(x^3 - p*q)

sage: K.maximal_order([3], assume_maximal=None).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

sage: K.maximal_order([2], assume_maximal=None).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

sage: K.maximal_order([p], assume_maximal=None).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

sage: K.maximal_order([q], assume_maximal=None).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

sage: K.maximal_order([p, 3], assume_maximal=None).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

Python

>>> from sage.all import *
>>> p = next_prime(Integer(10)**Integer(22))
>>> q = next_prime(Integer(10)**Integer(23))
>>> K = NumberField(x**Integer(3) - p*q, names=('a',)); (a,) = K._first_ngens(1)

>>> K.maximal_order([Integer(3)], assume_maximal=None).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

>>> K.maximal_order([Integer(2)], assume_maximal=None).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

>>> K.maximal_order([p], assume_maximal=None).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

>>> K.maximal_order([q], assume_maximal=None).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

>>> K.maximal_order([p, Integer(3)], assume_maximal=None).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]

An example with bigger discriminant:

Sage

sage: p = next_prime(10^97)
sage: q = next_prime(10^99)
sage: K.<a> = NumberField(x^3 - p*q)
sage: K.maximal_order(prime_range(10000), assume_maximal=None).basis()
[1, a, a^2]

Python

>>> from sage.all import *
>>> p = next_prime(Integer(10)**Integer(97))
>>> q = next_prime(Integer(10)**Integer(99))
>>> K = NumberField(x**Integer(3) - p*q, names=('a',)); (a,) = K._first_ngens(1)
>>> K.maximal_order(prime_range(Integer(10000)), assume_maximal=None).basis()
[1, a, a^2]

An example in a relative number field:

Sage

sage: K.<a, b> = NumberField([x^2 + 1, x^2 - 3])
sage: OK = K.maximal_order()
sage: OK.basis()
[1, 1/2*a - 1/2*b, -1/2*b*a + 1/2, a]

sage: charpoly(OK.1)
x^2 + b*x + 1
sage: charpoly(OK.2)
x^2 - x + 1

sage: O2 = K.order([3*a, 2*b])
sage: O2.index_in(OK)
144

Python

>>> from sage.all import *
>>> K = NumberField([x**Integer(2) + Integer(1), x**Integer(2) - Integer(3)], names=('a', 'b',)); (a, b,) = K._first_ngens(2)
>>> OK = K.maximal_order()
>>> OK.basis()
[1, 1/2*a - 1/2*b, -1/2*b*a + 1/2, a]

>>> charpoly(OK.gen(1))
x^2 + b*x + 1
>>> charpoly(OK.gen(2))
x^2 - x + 1

>>> O2 = K.order([Integer(3)*a, Integer(2)*b])
>>> O2.index_in(OK)
144

An order that is maximal at a prime. We happen to know that it is actually maximal and mark it as such:

Sage

sage: K.<i> = NumberField(x^2 + 1)
sage: K.maximal_order(v=2, assume_maximal=True)
Gaussian Integers generated by i in Number Field in i with defining polynomial x^2 + 1

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.maximal_order(v=Integer(2), assume_maximal=True)
Gaussian Integers generated by i in Number Field in i with defining polynomial x^2 + 1

It is an error to create a maximal order and declare it non-maximal, however, such mistakes are only caught automatically if they evidently contradict previous results in this session:

Sage

sage: K.maximal_order(v=2, assume_maximal=False)
Traceback (most recent call last):
...
ValueError: cannot assume this order to be non-maximal
because we already found it to be a maximal order

Python

>>> from sage.all import *
>>> K.maximal_order(v=Integer(2), assume_maximal=False)
Traceback (most recent call last):
...
ValueError: cannot assume this order to be non-maximal
because we already found it to be a maximal order

maximal_totally_real_subfield()[source]¶

Return the maximal totally real subfield of self together with an embedding of it into self.

EXAMPLES:

Sage

sage: F.<a> = QuadraticField(11)
sage: F.maximal_totally_real_subfield()
[Number Field in a with defining polynomial x^2 - 11 with a = 3.316624790355400?,
 Identity endomorphism of
  Number Field in a with defining polynomial x^2 - 11 with a = 3.316624790355400?]
sage: F.<a> = QuadraticField(-15)
sage: F.maximal_totally_real_subfield()
[Rational Field, Natural morphism:
   From: Rational Field
   To:   Number Field in a with defining polynomial x^2 + 15
         with a = 3.872983346207417?*I]
sage: F.<a> = CyclotomicField(29)
sage: F.maximal_totally_real_subfield()
(Number Field in a0 with defining polynomial x^14 + x^13 - 13*x^12 - 12*x^11
   + 66*x^10 + 55*x^9 - 165*x^8 - 120*x^7 + 210*x^6 + 126*x^5 - 126*x^4
   - 56*x^3 + 28*x^2 + 7*x - 1 with a0 = 1.953241111420174?,
 Ring morphism:
   From: Number Field in a0 with defining polynomial x^14 + x^13 - 13*x^12 - 12*x^11
         + 66*x^10 + 55*x^9 - 165*x^8 - 120*x^7 + 210*x^6 + 126*x^5 - 126*x^4
         - 56*x^3 + 28*x^2 + 7*x - 1 with a0 = 1.953241111420174?
   To:   Cyclotomic Field of order 29 and degree 28
   Defn: a0 |--> -a^27 - a^26 - a^25 - a^24 - a^23 - a^22 - a^21 - a^20 - a^19
                 - a^18 - a^17 - a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10
                 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 - 1)
sage: x = polygen(QQ, 'x')
sage: F.<a> = NumberField(x^3 - 2)
sage: F.maximal_totally_real_subfield()
[Rational Field,
 Coercion map:
   From: Rational Field
   To:   Number Field in a with defining polynomial x^3 - 2]
sage: F.<a> = NumberField(x^4 - x^3 - x^2 + x + 1)
sage: F.maximal_totally_real_subfield()
[Rational Field,
 Coercion map:
   From: Rational Field
   To:   Number Field in a with defining polynomial x^4 - x^3 - x^2 + x + 1]
sage: F.<a> = NumberField(x^4 - x^3 + 2*x^2 + x + 1)
sage: F.maximal_totally_real_subfield()
[Number Field in a1 with defining polynomial x^2 - x - 1,
 Ring morphism:
  From: Number Field in a1 with defining polynomial x^2 - x - 1
  To:   Number Field in a with defining polynomial x^4 - x^3 + 2*x^2 + x + 1
  Defn: a1 |--> -1/2*a^3 - 1/2]
sage: F.<a> = NumberField(x^4 - 4*x^2 - x + 1)
sage: F.maximal_totally_real_subfield()
[Number Field in a with defining polynomial x^4 - 4*x^2 - x + 1,
 Identity endomorphism of
  Number Field in a with defining polynomial x^4 - 4*x^2 - x + 1]

Python

>>> from sage.all import *
>>> F = QuadraticField(Integer(11), names=('a',)); (a,) = F._first_ngens(1)
>>> F.maximal_totally_real_subfield()
[Number Field in a with defining polynomial x^2 - 11 with a = 3.316624790355400?,
 Identity endomorphism of
  Number Field in a with defining polynomial x^2 - 11 with a = 3.316624790355400?]
>>> F = QuadraticField(-Integer(15), names=('a',)); (a,) = F._first_ngens(1)
>>> F.maximal_totally_real_subfield()
[Rational Field, Natural morphism:
   From: Rational Field
   To:   Number Field in a with defining polynomial x^2 + 15
         with a = 3.872983346207417?*I]
>>> F = CyclotomicField(Integer(29), names=('a',)); (a,) = F._first_ngens(1)
>>> F.maximal_totally_real_subfield()
(Number Field in a0 with defining polynomial x^14 + x^13 - 13*x^12 - 12*x^11
   + 66*x^10 + 55*x^9 - 165*x^8 - 120*x^7 + 210*x^6 + 126*x^5 - 126*x^4
   - 56*x^3 + 28*x^2 + 7*x - 1 with a0 = 1.953241111420174?,
 Ring morphism:
   From: Number Field in a0 with defining polynomial x^14 + x^13 - 13*x^12 - 12*x^11
         + 66*x^10 + 55*x^9 - 165*x^8 - 120*x^7 + 210*x^6 + 126*x^5 - 126*x^4
         - 56*x^3 + 28*x^2 + 7*x - 1 with a0 = 1.953241111420174?
   To:   Cyclotomic Field of order 29 and degree 28
   Defn: a0 |--> -a^27 - a^26 - a^25 - a^24 - a^23 - a^22 - a^21 - a^20 - a^19
                 - a^18 - a^17 - a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10
                 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 - 1)
>>> x = polygen(QQ, 'x')
>>> F = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = F._first_ngens(1)
>>> F.maximal_totally_real_subfield()
[Rational Field,
 Coercion map:
   From: Rational Field
   To:   Number Field in a with defining polynomial x^3 - 2]
>>> F = NumberField(x**Integer(4) - x**Integer(3) - x**Integer(2) + x + Integer(1), names=('a',)); (a,) = F._first_ngens(1)
>>> F.maximal_totally_real_subfield()
[Rational Field,
 Coercion map:
   From: Rational Field
   To:   Number Field in a with defining polynomial x^4 - x^3 - x^2 + x + 1]
>>> F = NumberField(x**Integer(4) - x**Integer(3) + Integer(2)*x**Integer(2) + x + Integer(1), names=('a',)); (a,) = F._first_ngens(1)
>>> F.maximal_totally_real_subfield()
[Number Field in a1 with defining polynomial x^2 - x - 1,
 Ring morphism:
  From: Number Field in a1 with defining polynomial x^2 - x - 1
  To:   Number Field in a with defining polynomial x^4 - x^3 + 2*x^2 + x + 1
  Defn: a1 |--> -1/2*a^3 - 1/2]
>>> F = NumberField(x**Integer(4) - Integer(4)*x**Integer(2) - x + Integer(1), names=('a',)); (a,) = F._first_ngens(1)
>>> F.maximal_totally_real_subfield()
[Number Field in a with defining polynomial x^4 - 4*x^2 - x + 1,
 Identity endomorphism of
  Number Field in a with defining polynomial x^4 - 4*x^2 - x + 1]

An example of a relative extension where the base field is not the maximal totally real subfield.

Sage

sage: E_0.<a> = NumberField(x^2 - 4*x + 16)
sage: y = polygen(E_0)
sage: E.<z> = E_0.extension(y^2 - E_0.gen() / 2)
sage: E.maximal_totally_real_subfield()
[Number Field in z1 with defining polynomial x^2 - 2*x - 5,
 Composite map:
   From: Number Field in z1 with defining polynomial x^2 - 2*x - 5
   To:   Number Field in z with defining polynomial x^2 - 1/2*a over its base field
   Defn:   Ring morphism:
           From: Number Field in z1 with defining polynomial x^2 - 2*x - 5
           To:   Number Field in z with defining
                 polynomial x^4 - 2*x^3 + x^2 + 6*x + 3
           Defn: z1 |--> -1/3*z^3 + 1/3*z^2 + z - 1
         then
           Isomorphism map:
           From: Number Field in z with defining
                 polynomial x^4 - 2*x^3 + x^2 + 6*x + 3
           To:   Number Field in z with defining
                 polynomial x^2 - 1/2*a over its base field]

Python

>>> from sage.all import *
>>> E_0 = NumberField(x**Integer(2) - Integer(4)*x + Integer(16), names=('a',)); (a,) = E_0._first_ngens(1)
>>> y = polygen(E_0)
>>> E = E_0.extension(y**Integer(2) - E_0.gen() / Integer(2), names=('z',)); (z,) = E._first_ngens(1)
>>> E.maximal_totally_real_subfield()
[Number Field in z1 with defining polynomial x^2 - 2*x - 5,
 Composite map:
   From: Number Field in z1 with defining polynomial x^2 - 2*x - 5
   To:   Number Field in z with defining polynomial x^2 - 1/2*a over its base field
   Defn:   Ring morphism:
           From: Number Field in z1 with defining polynomial x^2 - 2*x - 5
           To:   Number Field in z with defining
                 polynomial x^4 - 2*x^3 + x^2 + 6*x + 3
           Defn: z1 |--> -1/3*z^3 + 1/3*z^2 + z - 1
         then
           Isomorphism map:
           From: Number Field in z with defining
                 polynomial x^4 - 2*x^3 + x^2 + 6*x + 3
           To:   Number Field in z with defining
                 polynomial x^2 - 1/2*a over its base field]

narrow_class_group(proof=None)[source]¶

Return the narrow class group of this field.

INPUT:

proof – (default: None) use the global proof setting, which defaults to True

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^3 + x + 9, 'a').narrow_class_group()
Multiplicative Abelian group isomorphic to C2

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(3) + x + Integer(9), 'a').narrow_class_group()
Multiplicative Abelian group isomorphic to C2

ngens()[source]¶

Return the number of generators of this number field (always 1).

OUTPUT: the python integer 1

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + 17,'a').ngens()
1
sage: NumberField(x + 3,'a').ngens()
1
sage: k.<a> = NumberField(x + 3)
sage: k.ngens()
1
sage: k.0
-3

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + Integer(17),'a').ngens()
1
>>> NumberField(x + Integer(3),'a').ngens()
1
>>> k = NumberField(x + Integer(3), names=('a',)); (a,) = k._first_ngens(1)
>>> k.ngens()
1
>>> k.gen(0)
-3

number_of_roots_of_unity()[source]¶

Return the number of roots of unity in this field.

Note

We do not create the full unit group since that can be expensive, but we do use it if it is already known.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: F.<alpha> = NumberField(x^22 + 3)
sage: F.zeta_order()
6
sage: F.<alpha> = NumberField(x^2 - 7)
sage: F.zeta_order()
2

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> F = NumberField(x**Integer(22) + Integer(3), names=('alpha',)); (alpha,) = F._first_ngens(1)
>>> F.zeta_order()
6
>>> F = NumberField(x**Integer(2) - Integer(7), names=('alpha',)); (alpha,) = F._first_ngens(1)
>>> F.zeta_order()
2

order()[source]¶

Return the order of this number field (always +infinity).

OUTPUT: always positive infinity

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + 19,'a').order()
+Infinity

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + Integer(19),'a').order()
+Infinity

pari_bnf(proof=None, units=True)[source]¶

PARI big number field corresponding to this field.

INPUT:

proof – if False, assume GRH; if True, run PARI’s pari:bnfcertify to make sure that the results are correct
units – (default: True) if ``True, insist on having fundamental units; if False, the units may or may not be computed

OUTPUT: the PARI bnf structure of this number field

Warning

Even with proof=True, I wouldn’t trust this to mean that everything computed involving this number field is actually correct.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 1); k
Number Field in a with defining polynomial x^2 + 1
sage: len(k.pari_bnf())
10
sage: k.pari_bnf()[:4]
[[;], matrix(0,3), [;], ...]
sage: len(k.pari_nf())
9
sage: k.<a> = NumberField(x^7 + 7); k
Number Field in a with defining polynomial x^7 + 7
sage: dummy = k.pari_bnf(proof=True)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = k._first_ngens(1); k
Number Field in a with defining polynomial x^2 + 1
>>> len(k.pari_bnf())
10
>>> k.pari_bnf()[:Integer(4)]
[[;], matrix(0,3), [;], ...]
>>> len(k.pari_nf())
9
>>> k = NumberField(x**Integer(7) + Integer(7), names=('a',)); (a,) = k._first_ngens(1); k
Number Field in a with defining polynomial x^7 + 7
>>> dummy = k.pari_bnf(proof=True)

pari_nf(important=True)[source]¶

Return the PARI number field corresponding to this field.

INPUT:

important – boolean (default: True); if False, raise a RuntimeError if we need to do a difficult discriminant factorization. This is useful when an integral basis is not strictly required, such as for factoring polynomials over this number field.

OUTPUT:

The PARI number field obtained by calling the PARI function pari:nfinit with self.pari_polynomial('y') as argument.

Note

This method has the same effect as pari(self).

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^4 - 3*x + 7); k
Number Field in a with defining polynomial x^4 - 3*x + 7
sage: k.pari_nf()[:4]
[y^4 - 3*y + 7, [0, 2], 85621, 1]
sage: pari(k)[:4]
[y^4 - 3*y + 7, [0, 2], 85621, 1]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(4) - Integer(3)*x + Integer(7), names=('a',)); (a,) = k._first_ngens(1); k
Number Field in a with defining polynomial x^4 - 3*x + 7
>>> k.pari_nf()[:Integer(4)]
[y^4 - 3*y + 7, [0, 2], 85621, 1]
>>> pari(k)[:Integer(4)]
[y^4 - 3*y + 7, [0, 2], 85621, 1]

Sage

sage: k.<a> = NumberField(x^4 - 3/2*x + 5/3); k
Number Field in a with defining polynomial x^4 - 3/2*x + 5/3
sage: k.pari_nf()
[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ..., [36, 36*y, y^3 + 6*y^2 - 252, 6*y^2], [1, 0, 0, 252; 0, 1, 0, 0; 0, 0, 0, 36; 0, 0, 6, -36], [1, 0, 0, 0, 0, 0, -18, 42, 0, -18, -46, -60, 0, 42, -60, -60; 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, -11, -1, 0, 0, -1, 9; 0, 0, 1, 0, 0, 0, 6, 6, 1, 6, -5, 0, 0, 6, 0, 0; 0, 0, 0, 1, 0, 6, -6, -6, 0, -6, -1, 2, 1, -6, 2, 0]]
sage: pari(k)
[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ...]
sage: gp(k)
[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ...]

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(4) - Integer(3)/Integer(2)*x + Integer(5)/Integer(3), names=('a',)); (a,) = k._first_ngens(1); k
Number Field in a with defining polynomial x^4 - 3/2*x + 5/3
>>> k.pari_nf()
[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ..., [36, 36*y, y^3 + 6*y^2 - 252, 6*y^2], [1, 0, 0, 252; 0, 1, 0, 0; 0, 0, 0, 36; 0, 0, 6, -36], [1, 0, 0, 0, 0, 0, -18, 42, 0, -18, -46, -60, 0, 42, -60, -60; 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, -11, -1, 0, 0, -1, 9; 0, 0, 1, 0, 0, 0, 6, 6, 1, 6, -5, 0, 0, 6, 0, 0; 0, 0, 0, 1, 0, 6, -6, -6, 0, -6, -1, 2, 1, -6, 2, 0]]
>>> pari(k)
[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ...]
>>> gp(k)
[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ...]

With important=False, we simply bail out if we cannot easily factor the discriminant:

Sage

sage: p = next_prime(10^40); q = next_prime(10^41)
sage: K.<a> = NumberField(x^2 - p*q)
sage: K.pari_nf(important=False)
Traceback (most recent call last):
...
RuntimeError: Unable to factor discriminant with trial division

Python

>>> from sage.all import *
>>> p = next_prime(Integer(10)**Integer(40)); q = next_prime(Integer(10)**Integer(41))
>>> K = NumberField(x**Integer(2) - p*q, names=('a',)); (a,) = K._first_ngens(1)
>>> K.pari_nf(important=False)
Traceback (most recent call last):
...
RuntimeError: Unable to factor discriminant with trial division

Next, we illustrate the maximize_at_primes and assume_disc_small parameters of the NumberField constructor. The following would take a very long time without the maximize_at_primes option:

Sage

sage: K.<a> = NumberField(x^2 - p*q, maximize_at_primes=[p])
sage: K.pari_nf()
[y^2 - 100000000000000000000...]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) - p*q, maximize_at_primes=[p], names=('a',)); (a,) = K._first_ngens(1)
>>> K.pari_nf()
[y^2 - 100000000000000000000...]

Since the discriminant is square-free, this also works:

Sage

sage: K.<a> = NumberField(x^2 - p*q, assume_disc_small=True)
sage: K.pari_nf()
[y^2 - 100000000000000000000...]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) - p*q, assume_disc_small=True, names=('a',)); (a,) = K._first_ngens(1)
>>> K.pari_nf()
[y^2 - 100000000000000000000...]

pari_polynomial(name='x')[source]¶

Return the PARI polynomial corresponding to this number field.

INPUT:

name – variable name (default: 'x')

OUTPUT:

A monic polynomial with integral coefficients (PARI t_POL) defining the PARI number field corresponding to self.

Warning

This is not the same as simply converting the defining polynomial to PARI.

EXAMPLES:

Sage

sage: y = polygen(QQ)
sage: k.<a> = NumberField(y^2 - 3/2*y + 5/3)
sage: k.pari_polynomial()
x^2 - x + 40
sage: k.polynomial().__pari__()
x^2 - 3/2*x + 5/3
sage: k.pari_polynomial('a')
a^2 - a + 40

Python

>>> from sage.all import *
>>> y = polygen(QQ)
>>> k = NumberField(y**Integer(2) - Integer(3)/Integer(2)*y + Integer(5)/Integer(3), names=('a',)); (a,) = k._first_ngens(1)
>>> k.pari_polynomial()
x^2 - x + 40
>>> k.polynomial().__pari__()
x^2 - 3/2*x + 5/3
>>> k.pari_polynomial('a')
a^2 - a + 40

Some examples with relative number fields:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a, c> = NumberField([x^2 + 3, x^2 + 1])
sage: k.pari_polynomial()
x^4 + 8*x^2 + 4
sage: k.pari_polynomial('a')
a^4 + 8*a^2 + 4
sage: k.absolute_polynomial()
x^4 + 8*x^2 + 4
sage: k.relative_polynomial()
x^2 + 3

sage: k.<a, c> = NumberField([x^2 + 1/3, x^2 + 1/4])
sage: k.pari_polynomial()
x^4 - x^2 + 1
sage: k.absolute_polynomial()
x^4 - x^2 + 1

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField([x**Integer(2) + Integer(3), x**Integer(2) + Integer(1)], names=('a', 'c',)); (a, c,) = k._first_ngens(2)
>>> k.pari_polynomial()
x^4 + 8*x^2 + 4
>>> k.pari_polynomial('a')
a^4 + 8*a^2 + 4
>>> k.absolute_polynomial()
x^4 + 8*x^2 + 4
>>> k.relative_polynomial()
x^2 + 3

>>> k = NumberField([x**Integer(2) + Integer(1)/Integer(3), x**Integer(2) + Integer(1)/Integer(4)], names=('a', 'c',)); (a, c,) = k._first_ngens(2)
>>> k.pari_polynomial()
x^4 - x^2 + 1
>>> k.absolute_polynomial()
x^4 - x^2 + 1

This fails with arguments which are not a valid PARI variable name:

Sage

sage: k = QuadraticField(-1)
sage: k.pari_polynomial('I')
Traceback (most recent call last):
...
PariError: I already exists with incompatible valence
sage: k.pari_polynomial('i')
i^2 + 1
sage: k.pari_polynomial('theta')
Traceback (most recent call last):
...
PariError: theta already exists with incompatible valence

Python

>>> from sage.all import *
>>> k = QuadraticField(-Integer(1))
>>> k.pari_polynomial('I')
Traceback (most recent call last):
...
PariError: I already exists with incompatible valence
>>> k.pari_polynomial('i')
i^2 + 1
>>> k.pari_polynomial('theta')
Traceback (most recent call last):
...
PariError: theta already exists with incompatible valence

pari_rnfnorm_data(L, proof=True)[source]¶

Return the PARI pari:rnfisnorminit data corresponding to the extension \(L\) / self.

EXAMPLES:

Sage

sage: x = polygen(QQ)
sage: K = NumberField(x^2 - 2, 'alpha')
sage: L = K.extension(x^2 + 5, 'gamma')
sage: ls = K.pari_rnfnorm_data(L) ; len(ls)
8

sage: K.<a> = NumberField(x^2 + x + 1)
sage: P.<X> = K[]
sage: L.<b> = NumberField(X^3 + a)
sage: ls = K.pari_rnfnorm_data(L); len(ls)
8

Python

>>> from sage.all import *
>>> x = polygen(QQ)
>>> K = NumberField(x**Integer(2) - Integer(2), 'alpha')
>>> L = K.extension(x**Integer(2) + Integer(5), 'gamma')
>>> ls = K.pari_rnfnorm_data(L) ; len(ls)
8

>>> K = NumberField(x**Integer(2) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K['X']; (X,) = P._first_ngens(1)
>>> L = NumberField(X**Integer(3) + a, names=('b',)); (b,) = L._first_ngens(1)
>>> ls = K.pari_rnfnorm_data(L); len(ls)
8

pari_zk()[source]¶

Integral basis of the PARI number field corresponding to this field.

This is the same as pari_nf().getattr('zk'), but much faster.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^3 - 17)
sage: k.pari_zk()
[1, 1/3*y^2 - 1/3*y + 1/3, y]
sage: k.pari_nf().getattr('zk')
[1, 1/3*y^2 - 1/3*y + 1/3, y]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(3) - Integer(17), names=('a',)); (a,) = k._first_ngens(1)
>>> k.pari_zk()
[1, 1/3*y^2 - 1/3*y + 1/3, y]
>>> k.pari_nf().getattr('zk')
[1, 1/3*y^2 - 1/3*y + 1/3, y]

polynomial()[source]¶

Return the defining polynomial of this number field.

This is exactly the same as defining_polynomial().

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial()
x^2 + 2/3*x - 9/17

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + (Integer(2)/Integer(3))*x - Integer(9)/Integer(17),'a').polynomial()
x^2 + 2/3*x - 9/17

polynomial_ntl()[source]¶

Return defining polynomial of this number field as a pair, an ntl polynomial and a denominator.

This is used mainly to implement some internal arithmetic.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial_ntl()
([-27 34 51], 51)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + (Integer(2)/Integer(3))*x - Integer(9)/Integer(17),'a').polynomial_ntl()
([-27 34 51], 51)

polynomial_quotient_ring()[source]¶

Return the polynomial quotient ring isomorphic to this number field.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K = NumberField(x^3 + 2*x - 5, 'alpha')
sage: K.polynomial_quotient_ring()
Univariate Quotient Polynomial Ring in alpha over Rational Field
 with modulus x^3 + 2*x - 5

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) + Integer(2)*x - Integer(5), 'alpha')
>>> K.polynomial_quotient_ring()
Univariate Quotient Polynomial Ring in alpha over Rational Field
 with modulus x^3 + 2*x - 5

polynomial_ring()[source]¶

Return the polynomial ring that we view this number field as being a quotient of (by a principal ideal).

EXAMPLES: An example with an absolute field:

Sage

sage: x = polygen(QQ, 'x')
sage: k.<a> = NumberField(x^2 + 3)
sage: y = polygen(QQ, 'y')
sage: k.<a> = NumberField(y^2 + 3)
sage: k.polynomial_ring()
Univariate Polynomial Ring in y over Rational Field

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(3), names=('a',)); (a,) = k._first_ngens(1)
>>> y = polygen(QQ, 'y')
>>> k = NumberField(y**Integer(2) + Integer(3), names=('a',)); (a,) = k._first_ngens(1)
>>> k.polynomial_ring()
Univariate Polynomial Ring in y over Rational Field

An example with a relative field:

Sage

sage: y = polygen(QQ, 'y')
sage: M.<a> = NumberField([y^3 + 97, y^2 + 1]); M
Number Field in a0 with defining polynomial y^3 + 97 over its base field
sage: M.polynomial_ring()
Univariate Polynomial Ring in y over
 Number Field in a1 with defining polynomial y^2 + 1

Python

>>> from sage.all import *
>>> y = polygen(QQ, 'y')
>>> M = NumberField([y**Integer(3) + Integer(97), y**Integer(2) + Integer(1)], names=('a',)); (a,) = M._first_ngens(1); M
Number Field in a0 with defining polynomial y^3 + 97 over its base field
>>> M.polynomial_ring()
Univariate Polynomial Ring in y over
 Number Field in a1 with defining polynomial y^2 + 1

power_basis()[source]¶

Return a power basis for this number field over its base field.

If this number field is represented as \(k[t]/f(t)\), then the basis returned is \(1, t, t^2, \ldots, t^{d-1}\) where \(d\) is the degree of this number field over its base field.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^5 + 10*x + 1)
sage: K.power_basis()
[1, a, a^2, a^3, a^4]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(5) + Integer(10)*x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.power_basis()
[1, a, a^2, a^3, a^4]

Sage

sage: L.<b> = K.extension(x^2 - 2)
sage: L.power_basis()
[1, b]
sage: L.absolute_field('c').power_basis()
[1, c, c^2, c^3, c^4, c^5, c^6, c^7, c^8, c^9]

Python

>>> from sage.all import *
>>> L = K.extension(x**Integer(2) - Integer(2), names=('b',)); (b,) = L._first_ngens(1)
>>> L.power_basis()
[1, b]
>>> L.absolute_field('c').power_basis()
[1, c, c^2, c^3, c^4, c^5, c^6, c^7, c^8, c^9]

Sage

sage: M = CyclotomicField(15)
sage: M.power_basis()
[1, zeta15, zeta15^2, zeta15^3, zeta15^4, zeta15^5, zeta15^6, zeta15^7]

Python

>>> from sage.all import *
>>> M = CyclotomicField(Integer(15))
>>> M.power_basis()
[1, zeta15, zeta15^2, zeta15^3, zeta15^4, zeta15^5, zeta15^6, zeta15^7]

prime_above(x, degree=None)[source]¶

Return a prime ideal of self lying over \(x\).

INPUT:

x – usually an element or ideal of self. It should be such that self.ideal(x) is sensible. This excludes \(x=0\).
degree – (default: None) None or an integer. If one, find a prime above \(x\) of any degree. If an integer, find a prime above \(x\) such that the resulting residue field has exactly this degree.

OUTPUT: a prime ideal of self lying over \(x\). If degree is specified and no such ideal exists, raises a ValueError.

EXAMPLES:

Sage

sage: x = ZZ['x'].gen()
sage: F.<t> = NumberField(x^3 - 2)

Python

>>> from sage.all import *
>>> x = ZZ['x'].gen()
>>> F = NumberField(x**Integer(3) - Integer(2), names=('t',)); (t,) = F._first_ngens(1)

Sage

sage: P2 = F.prime_above(2)
sage: P2 # random
Fractional ideal (-t)
sage: 2 in P2
True
sage: P2.is_prime()
True
sage: P2.norm()
2

Python

>>> from sage.all import *
>>> P2 = F.prime_above(Integer(2))
>>> P2 # random
Fractional ideal (-t)
>>> Integer(2) in P2
True
>>> P2.is_prime()
True
>>> P2.norm()
2

Sage

sage: P3 = F.prime_above(3)
sage: P3 # random
Fractional ideal (t + 1)
sage: 3 in P3
True
sage: P3.is_prime()
True
sage: P3.norm()
3

Python

>>> from sage.all import *
>>> P3 = F.prime_above(Integer(3))
>>> P3 # random
Fractional ideal (t + 1)
>>> Integer(3) in P3
True
>>> P3.is_prime()
True
>>> P3.norm()
3

The ideal \((3)\) is totally ramified in \(F\), so there is no degree 2 prime above \(3\):

Sage

sage: F.prime_above(3, degree=2)
Traceback (most recent call last):
...
ValueError: No prime of degree 2 above Fractional ideal (3)
sage: [ id.residue_class_degree() for id, _ in F.ideal(3).factor() ]
[1]

Python

>>> from sage.all import *
>>> F.prime_above(Integer(3), degree=Integer(2))
Traceback (most recent call last):
...
ValueError: No prime of degree 2 above Fractional ideal (3)
>>> [ id.residue_class_degree() for id, _ in F.ideal(Integer(3)).factor() ]
[1]

Asking for a specific degree works:

Sage

sage: P5_1 = F.prime_above(5, degree=1)
sage: P5_1 # random
Fractional ideal (-t^2 - 1)
sage: P5_1.residue_class_degree()
1

Python

>>> from sage.all import *
>>> P5_1 = F.prime_above(Integer(5), degree=Integer(1))
>>> P5_1 # random
Fractional ideal (-t^2 - 1)
>>> P5_1.residue_class_degree()
1

Sage

sage: P5_2 = F.prime_above(5, degree=2)
sage: P5_2 # random
Fractional ideal (t^2 - 2*t - 1)
sage: P5_2.residue_class_degree()
2

Python

>>> from sage.all import *
>>> P5_2 = F.prime_above(Integer(5), degree=Integer(2))
>>> P5_2 # random
Fractional ideal (t^2 - 2*t - 1)
>>> P5_2.residue_class_degree()
2

Relative number fields are ok:

Sage

sage: G = F.extension(x^2 - 11, 'b')
sage: G.prime_above(7)
Fractional ideal (b + 2)

Python

>>> from sage.all import *
>>> G = F.extension(x**Integer(2) - Integer(11), 'b')
>>> G.prime_above(Integer(7))
Fractional ideal (b + 2)

It doesn’t make sense to factor the ideal \((0)\):

Sage

sage: F.prime_above(0)
Traceback (most recent call last):
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'prime_factors'...

Python

>>> from sage.all import *
>>> F.prime_above(Integer(0))
Traceback (most recent call last):
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'prime_factors'...

prime_factors(x)[source]¶

Return a list of the prime ideals of self which divide the ideal generated by \(x\).

OUTPUT: list of prime ideals (a new list is returned each time this function is called)

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<w> = NumberField(x^2 + 23)
sage: K.prime_factors(w + 1)
[Fractional ideal (2, 1/2*w - 1/2),
 Fractional ideal (2, 1/2*w + 1/2),
 Fractional ideal (3, 1/2*w + 1/2)]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(23), names=('w',)); (w,) = K._first_ngens(1)
>>> K.prime_factors(w + Integer(1))
[Fractional ideal (2, 1/2*w - 1/2),
 Fractional ideal (2, 1/2*w + 1/2),
 Fractional ideal (3, 1/2*w + 1/2)]

primes_above(x, degree=None)[source]¶

Return prime ideals of self lying over \(x\).

INPUT:

x – usually an element or ideal of self. It should be such that self.ideal(x) is sensible. This excludes \(x=0\).
degree – (default: None) None or an integer. If None, find all primes above \(x\) of any degree. If an integer, find all primes above \(x\) such that the resulting residue field has exactly this degree.

OUTPUT: list of prime ideals of self lying over \(x\). If degree is specified and no such ideal exists, returns the empty list. The output is sorted by residue degree first, then by underlying prime (or equivalently, by norm).

EXAMPLES:

Sage

sage: x = ZZ['x'].gen()
sage: F.<t> = NumberField(x^3 - 2)

Python

>>> from sage.all import *
>>> x = ZZ['x'].gen()
>>> F = NumberField(x**Integer(3) - Integer(2), names=('t',)); (t,) = F._first_ngens(1)

Sage

sage: P2s = F.primes_above(2)
sage: P2s  # random
[Fractional ideal (-t)]
sage: all(2 in P2 for P2 in P2s)
True
sage: all(P2.is_prime() for P2 in P2s)
True
sage: [ P2.norm() for P2 in P2s ]
[2]

Python

>>> from sage.all import *
>>> P2s = F.primes_above(Integer(2))
>>> P2s  # random
[Fractional ideal (-t)]
>>> all(Integer(2) in P2 for P2 in P2s)
True
>>> all(P2.is_prime() for P2 in P2s)
True
>>> [ P2.norm() for P2 in P2s ]
[2]

Sage

sage: P3s = F.primes_above(3)
sage: P3s # random
[Fractional ideal (t + 1)]
sage: all(3 in P3 for P3 in P3s)
True
sage: all(P3.is_prime() for P3 in P3s)
True
sage: [ P3.norm() for P3 in P3s ]
[3]

Python

>>> from sage.all import *
>>> P3s = F.primes_above(Integer(3))
>>> P3s # random
[Fractional ideal (t + 1)]
>>> all(Integer(3) in P3 for P3 in P3s)
True
>>> all(P3.is_prime() for P3 in P3s)
True
>>> [ P3.norm() for P3 in P3s ]
[3]

The ideal \((3)\) is totally ramified in \(F\), so there is no degree 2 prime above 3:

Sage

sage: F.primes_above(3, degree=2)
[]
sage: [ id.residue_class_degree() for id, _ in F.ideal(3).factor() ]
[1]

Python

>>> from sage.all import *
>>> F.primes_above(Integer(3), degree=Integer(2))
[]
>>> [ id.residue_class_degree() for id, _ in F.ideal(Integer(3)).factor() ]
[1]

Asking for a specific degree works:

Sage

sage: P5_1s = F.primes_above(5, degree=1)
sage: P5_1s # random
[Fractional ideal (-t^2 - 1)]
sage: P5_1 = P5_1s[0]; P5_1.residue_class_degree()
1

Python

>>> from sage.all import *
>>> P5_1s = F.primes_above(Integer(5), degree=Integer(1))
>>> P5_1s # random
[Fractional ideal (-t^2 - 1)]
>>> P5_1 = P5_1s[Integer(0)]; P5_1.residue_class_degree()
1

Sage

sage: P5_2s = F.primes_above(5, degree=2)
sage: P5_2s # random
[Fractional ideal (t^2 - 2*t - 1)]
sage: P5_2 = P5_2s[0]; P5_2.residue_class_degree()
2

Python

>>> from sage.all import *
>>> P5_2s = F.primes_above(Integer(5), degree=Integer(2))
>>> P5_2s # random
[Fractional ideal (t^2 - 2*t - 1)]
>>> P5_2 = P5_2s[Integer(0)]; P5_2.residue_class_degree()
2

Works in relative extensions too:

Sage

sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: I = F.ideal(a + 2*b)
sage: P, Q = K.primes_above(I)
sage: K.ideal(I) == P^4*Q
True
sage: K.primes_above(I, degree=1) == [P]
True
sage: K.primes_above(I, degree=4) == [Q]
True

Python

>>> from sage.all import *
>>> PQ = QQ['X']; (X,) = PQ._first_ngens(1)
>>> F = NumberField([X**Integer(2) - Integer(2), X**Integer(2) - Integer(3)], names=('a', 'b',)); (a, b,) = F._first_ngens(2)
>>> PF = F['Y']; (Y,) = PF._first_ngens(1)
>>> K = F.extension(Y**Integer(2) - (Integer(1) + a)*(a + b)*a*b, names=('c',)); (c,) = K._first_ngens(1)
>>> I = F.ideal(a + Integer(2)*b)
>>> P, Q = K.primes_above(I)
>>> K.ideal(I) == P**Integer(4)*Q
True
>>> K.primes_above(I, degree=Integer(1)) == [P]
True
>>> K.primes_above(I, degree=Integer(4)) == [Q]
True

It doesn’t make sense to factor the ideal \((0)\), so this raises an error:

Sage

sage: F.prime_above(0)
Traceback (most recent call last):
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'prime_factors'...

Python

>>> from sage.all import *
>>> F.prime_above(Integer(0))
Traceback (most recent call last):
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'prime_factors'...

primes_of_bounded_norm(B)[source]¶

Return a sorted list of all prime ideals with norm at most \(B\).

INPUT:

B – positive integer or real; upper bound on the norms of the primes generated

OUTPUT:

A list of all prime ideals of this number field of norm at most \(B\), sorted by norm. Primes of the same norm are sorted using the comparison function for ideals, which is based on the Hermite Normal Form.

Note

See also primes_of_bounded_norm_iter() for an iterator version of this, but note that the iterator sorts the primes in order of underlying rational prime, not by norm.

EXAMPLES:

Sage

sage: K.<i> = QuadraticField(-1)
sage: K.primes_of_bounded_norm(10)
[Fractional ideal (i + 1), Fractional ideal (-i - 2),
 Fractional ideal (2*i + 1), Fractional ideal (3)]
sage: K.primes_of_bounded_norm(1)
[]
sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: P = K.primes_of_bounded_norm(30)
sage: P
[Fractional ideal (a),
 Fractional ideal (a + 1),
 Fractional ideal (-a^2 - 1),
 Fractional ideal (a^2 + a - 1),
 Fractional ideal (2*a + 1),
 Fractional ideal (-2*a^2 - a - 1),
 Fractional ideal (a^2 - 2*a - 1),
 Fractional ideal (a + 3)]
sage: [p.norm() for p in P]
[2, 3, 5, 11, 17, 23, 25, 29]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.primes_of_bounded_norm(Integer(10))
[Fractional ideal (i + 1), Fractional ideal (-i - 2),
 Fractional ideal (2*i + 1), Fractional ideal (3)]
>>> K.primes_of_bounded_norm(Integer(1))
[]
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.primes_of_bounded_norm(Integer(30))
>>> P
[Fractional ideal (a),
 Fractional ideal (a + 1),
 Fractional ideal (-a^2 - 1),
 Fractional ideal (a^2 + a - 1),
 Fractional ideal (2*a + 1),
 Fractional ideal (-2*a^2 - a - 1),
 Fractional ideal (a^2 - 2*a - 1),
 Fractional ideal (a + 3)]
>>> [p.norm() for p in P]
[2, 3, 5, 11, 17, 23, 25, 29]

primes_of_bounded_norm_iter(B)[source]¶

Iterator yielding all prime ideals with norm at most \(B\).

INPUT:

B – positive integer or real; upper bound on the norms of the primes generated

OUTPUT:

An iterator over all prime ideals of this number field of norm at most \(B\).

Note

The output is not sorted by norm, but by size of the underlying rational prime.

EXAMPLES:

Sage

sage: K.<i> = QuadraticField(-1)
sage: it = K.primes_of_bounded_norm_iter(10)
sage: list(it)
[Fractional ideal (i + 1),
 Fractional ideal (3),
 Fractional ideal (-i - 2),
 Fractional ideal (2*i + 1)]
sage: list(K.primes_of_bounded_norm_iter(1))
[]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> it = K.primes_of_bounded_norm_iter(Integer(10))
>>> list(it)
[Fractional ideal (i + 1),
 Fractional ideal (3),
 Fractional ideal (-i - 2),
 Fractional ideal (2*i + 1)]
>>> list(K.primes_of_bounded_norm_iter(Integer(1)))
[]

primes_of_degree_one_iter(num_integer_primes=10000, max_iterations=100)[source]¶

Return an iterator yielding prime ideals of absolute degree one and small norm.

Warning

It is possible that there are no primes of \(K\) of absolute degree one of small prime norm, and it possible that this algorithm will not find any primes of small norm.

See module sage.rings.number_field.small_primes_of_degree_one for details.

INPUT:

num_integer_primes – (default: 10000) an integer. We try to find primes of absolute norm no greater than the num_integer_primes-th prime number. For example, if num_integer_primes is 2, the largest norm found will be 3, since the second prime is 3.
max_iterations – (default: 100) an integer. We test max_iterations integers to find small primes before raising StopIteration.

EXAMPLES:

Sage

sage: K.<z> = CyclotomicField(10)
sage: it = K.primes_of_degree_one_iter()
sage: Ps = [ next(it) for i in range(3) ]
sage: Ps # random
[Fractional ideal (z^3 + z + 1),
 Fractional ideal (3*z^3 - z^2 + z - 1),
 Fractional ideal (2*z^3 - 3*z^2 + z - 2)]
sage: [P.norm() for P in Ps] # random
[11, 31, 41]
sage: [P.residue_class_degree() for P in Ps]
[1, 1, 1]

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(10), names=('z',)); (z,) = K._first_ngens(1)
>>> it = K.primes_of_degree_one_iter()
>>> Ps = [ next(it) for i in range(Integer(3)) ]
>>> Ps # random
[Fractional ideal (z^3 + z + 1),
 Fractional ideal (3*z^3 - z^2 + z - 1),
 Fractional ideal (2*z^3 - 3*z^2 + z - 2)]
>>> [P.norm() for P in Ps] # random
[11, 31, 41]
>>> [P.residue_class_degree() for P in Ps]
[1, 1, 1]

primes_of_degree_one_list(n, num_integer_primes=10000, max_iterations=100)[source]¶

Return a list of \(n\) prime ideals of absolute degree one and small norm.

Warning

It is possible that there are no primes of \(K\) of absolute degree one of small prime norm, and it is possible that this algorithm will not find any primes of small norm.

See module sage.rings.number_field.small_primes_of_degree_one for details.

INPUT:

num_integer_primes – integer (default: 10000). We try to find primes of absolute norm no greater than the num_integer_primes-th prime number. For example, if num_integer_primes is 2, the largest norm found will be 3, since the second prime is 3.
max_iterations – integer (default: 100). We test max_iterations integers to find small primes before raising StopIteration.

EXAMPLES:

Sage

sage: K.<z> = CyclotomicField(10)
sage: Ps = K.primes_of_degree_one_list(3)
sage: Ps  # random output
[Fractional ideal (-z^3 - z^2 + 1),
 Fractional ideal (2*z^3 - 2*z^2 + 2*z - 3),
 Fractional ideal (2*z^3 - 3*z^2 + z - 2)]
sage: [P.norm() for P in Ps]
[11, 31, 41]
sage: [P.residue_class_degree() for P in Ps]
[1, 1, 1]

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(10), names=('z',)); (z,) = K._first_ngens(1)
>>> Ps = K.primes_of_degree_one_list(Integer(3))
>>> Ps  # random output
[Fractional ideal (-z^3 - z^2 + 1),
 Fractional ideal (2*z^3 - 2*z^2 + 2*z - 3),
 Fractional ideal (2*z^3 - 3*z^2 + z - 2)]
>>> [P.norm() for P in Ps]
[11, 31, 41]
>>> [P.residue_class_degree() for P in Ps]
[1, 1, 1]

primitive_element()[source]¶

Return a primitive element for this field, i.e., an element that generates it over \(\QQ\).

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 + 2)
sage: K.primitive_element()
a
sage: K.<a,b,c> = NumberField([x^2 - 2, x^2 - 3, x^2 - 5])
sage: K.primitive_element()
a - b + c
sage: alpha = K.primitive_element(); alpha
a - b + c
sage: alpha.minpoly()
x^2 + (2*b - 2*c)*x - 2*c*b + 6
sage: alpha.absolute_minpoly()
x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.primitive_element()
a
>>> K = NumberField([x**Integer(2) - Integer(2), x**Integer(2) - Integer(3), x**Integer(2) - Integer(5)], names=('a', 'b', 'c',)); (a, b, c,) = K._first_ngens(3)
>>> K.primitive_element()
a - b + c
>>> alpha = K.primitive_element(); alpha
a - b + c
>>> alpha.minpoly()
x^2 + (2*b - 2*c)*x - 2*c*b + 6
>>> alpha.absolute_minpoly()
x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576

primitive_root_of_unity()[source]¶

Return a generator of the roots of unity in this field.

OUTPUT: a primitive root of unity. No guarantee is made about which primitive root of unity this returns, not even for cyclotomic fields. Repeated calls of this function may return a different value.

Note

We do not create the full unit group since that can be expensive, but we do use it if it is already known.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: z = K.primitive_root_of_unity(); z
i
sage: z.multiplicative_order()
4

sage: K.<a> = NumberField(x^2 + x + 1)
sage: z = K.primitive_root_of_unity(); z
a + 1
sage: z.multiplicative_order()
6

sage: x = polygen(QQ)
sage: F.<a,b> = NumberField([x^2 - 2, x^2 - 3])
sage: y = polygen(F)
sage: K.<c> = F.extension(y^2 - (1 + a)*(a + b)*a*b)
sage: K.primitive_root_of_unity()
-1

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> z = K.primitive_root_of_unity(); z
i
>>> z.multiplicative_order()
4

>>> K = NumberField(x**Integer(2) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> z = K.primitive_root_of_unity(); z
a + 1
>>> z.multiplicative_order()
6

>>> x = polygen(QQ)
>>> F = NumberField([x**Integer(2) - Integer(2), x**Integer(2) - Integer(3)], names=('a', 'b',)); (a, b,) = F._first_ngens(2)
>>> y = polygen(F)
>>> K = F.extension(y**Integer(2) - (Integer(1) + a)*(a + b)*a*b, names=('c',)); (c,) = K._first_ngens(1)
>>> K.primitive_root_of_unity()
-1

We do not special-case cyclotomic fields, so we do not always get the most obvious primitive root of unity:

Sage

sage: K.<a> = CyclotomicField(3)
sage: z = K.primitive_root_of_unity(); z
a + 1
sage: z.multiplicative_order()
6

sage: K = CyclotomicField(3)
sage: z = K.primitive_root_of_unity(); z
zeta3 + 1
sage: z.multiplicative_order()
6

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> z = K.primitive_root_of_unity(); z
a + 1
>>> z.multiplicative_order()
6

>>> K = CyclotomicField(Integer(3))
>>> z = K.primitive_root_of_unity(); z
zeta3 + 1
>>> z.multiplicative_order()
6

quadratic_defect(a, p, check=True)[source]¶

Return the valuation of the quadratic defect of \(a\) at \(p\).

INPUT:

a – an element of self
p – a prime ideal
check – boolean (default: True); check if \(p\) is prime

ALGORITHM:

This is an implementation of Algorithm 3.1.3 from [Kir2016].

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 + 2)
sage: p = K.primes_above(2)[0]
sage: K.quadratic_defect(5, p)
4
sage: K.quadratic_defect(0, p)
+Infinity
sage: K.quadratic_defect(a, p)
1
sage: K.<a> = CyclotomicField(5)
sage: p = K.primes_above(2)[0]
sage: K.quadratic_defect(5, p)
+Infinity

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> p = K.primes_above(Integer(2))[Integer(0)]
>>> K.quadratic_defect(Integer(5), p)
4
>>> K.quadratic_defect(Integer(0), p)
+Infinity
>>> K.quadratic_defect(a, p)
1
>>> K = CyclotomicField(Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> p = K.primes_above(Integer(2))[Integer(0)]
>>> K.quadratic_defect(Integer(5), p)
+Infinity

random_element(num_bound=None, den_bound=None, integral_coefficients=False, distribution=None)[source]¶

Return a random element of this number field.

INPUT:

num_bound – bound on numerator of the coefficients of the resulting element
den_bound – bound on denominators of the coefficients of the resulting element
integral_coefficients – boolean (default: False); if True, then the resulting element will have integral coefficients. This option overrides any value of den_bound.
distribution – distribution to use for the coefficients of the resulting element

OUTPUT: element of this number field

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<j> = NumberField(x^8 + 1)
sage: K.random_element().parent() is K
True

sage: while K.random_element().list()[0] != 0:
....:     pass
sage: while K.random_element().list()[0] == 0:
....:     pass
sage: while K.random_element().is_prime():
....:     pass
sage: while not K.random_element().is_prime():
....:     pass

sage: K.<a,b,c> = NumberField([x^2 - 2, x^2 - 3, x^2 - 5])
sage: K.random_element().parent() is K
True

sage: while K.random_element().is_prime():
....:     pass
sage: while not K.random_element().is_prime():  # long time
....:     pass

sage: K.<a> = NumberField(x^5 - 2)
sage: p = K.random_element(integral_coefficients=True)
sage: p.is_integral()
True
sage: while K.random_element().is_integral():
....:     pass

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(8) + Integer(1), names=('j',)); (j,) = K._first_ngens(1)
>>> K.random_element().parent() is K
True

>>> while K.random_element().list()[Integer(0)] != Integer(0):
...     pass
>>> while K.random_element().list()[Integer(0)] == Integer(0):
...     pass
>>> while K.random_element().is_prime():
...     pass
>>> while not K.random_element().is_prime():
...     pass

>>> K = NumberField([x**Integer(2) - Integer(2), x**Integer(2) - Integer(3), x**Integer(2) - Integer(5)], names=('a', 'b', 'c',)); (a, b, c,) = K._first_ngens(3)
>>> K.random_element().parent() is K
True

>>> while K.random_element().is_prime():
...     pass
>>> while not K.random_element().is_prime():  # long time
...     pass

>>> K = NumberField(x**Integer(5) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> p = K.random_element(integral_coefficients=True)
>>> p.is_integral()
True
>>> while K.random_element().is_integral():
...     pass

real_embeddings(prec=53)[source]¶

Return all homomorphisms of this number field into the approximate real field with precision prec.

If prec is 53 (the default), then the real double field is used; otherwise the arbitrary precision (but slow) real field is used. If you want embeddings into the 53-bit double precision, which is faster, use self.embeddings(RDF).

Note

This function uses finite precision real numbers. In functions that should output proven results, one could use self.embeddings(AA) instead.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 + 2)
sage: K.real_embeddings()
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Real Field with 53 bits of precision
  Defn: a |--> -1.25992104989487
]
sage: K.real_embeddings(16)
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Real Field with 16 bits of precision
  Defn: a |--> -1.260
]
sage: K.real_embeddings(100)
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Real Field with 100 bits of precision
  Defn: a |--> -1.2599210498948731647672106073
]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) + Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.real_embeddings()
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Real Field with 53 bits of precision
  Defn: a |--> -1.25992104989487
]
>>> K.real_embeddings(Integer(16))
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Real Field with 16 bits of precision
  Defn: a |--> -1.260
]
>>> K.real_embeddings(Integer(100))
[
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + 2
  To:   Real Field with 100 bits of precision
  Defn: a |--> -1.2599210498948731647672106073
]

As this is a numerical function, the number of embeddings may be incorrect if the precision is too low:

Sage

sage: K = NumberField(x^2 + 2*10^1000*x + 10^2000 + 1, 'a')
sage: len(K.real_embeddings())
2
sage: len(K.real_embeddings(100))
2
sage: len(K.real_embeddings(10000))
0
sage: len(K.embeddings(AA))
0

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(2)*Integer(10)**Integer(1000)*x + Integer(10)**Integer(2000) + Integer(1), 'a')
>>> len(K.real_embeddings())
2
>>> len(K.real_embeddings(Integer(100)))
2
>>> len(K.real_embeddings(Integer(10000)))
0
>>> len(K.embeddings(AA))
0

reduced_basis(prec=None)[source]¶

Return an LLL-reduced basis for the Minkowski-embedding of the maximal order of a number field.

INPUT:

prec – (default: None) the precision with which to compute the Minkowski embedding

OUTPUT:

An LLL-reduced basis for the Minkowski-embedding of the maximal order of a number field, given by a sequence of (integral) elements from the field.

Note

In the non-totally-real case, the LLL routine we call is currently PARI’s pari:qflll, which works with floating point approximations, and so the result is only as good as the precision promised by PARI. The matrix returned will always be integral; however, it may only be only “almost” LLL-reduced when the precision is not sufficiently high.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: F.<t> = NumberField(x^6 - 7*x^4 - x^3 + 11*x^2 + x - 1)
sage: F.maximal_order().basis()
[1/2*t^5 + 1/2*t^4 + 1/2*t^2 + 1/2, t, t^2, t^3, t^4, t^5]
sage: F.reduced_basis()
[-1, -1/2*t^5 + 1/2*t^4 + 3*t^3 - 3/2*t^2 - 4*t - 1/2, t,
 1/2*t^5 + 1/2*t^4 - 4*t^3 - 5/2*t^2 + 7*t + 1/2,
 1/2*t^5 - 1/2*t^4 - 2*t^3 + 3/2*t^2 - 1/2,
 1/2*t^5 - 1/2*t^4 - 3*t^3 + 5/2*t^2 + 4*t - 5/2]
sage: CyclotomicField(12).reduced_basis()
[1, zeta12^2, zeta12, zeta12^3]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> F = NumberField(x**Integer(6) - Integer(7)*x**Integer(4) - x**Integer(3) + Integer(11)*x**Integer(2) + x - Integer(1), names=('t',)); (t,) = F._first_ngens(1)
>>> F.maximal_order().basis()
[1/2*t^5 + 1/2*t^4 + 1/2*t^2 + 1/2, t, t^2, t^3, t^4, t^5]
>>> F.reduced_basis()
[-1, -1/2*t^5 + 1/2*t^4 + 3*t^3 - 3/2*t^2 - 4*t - 1/2, t,
 1/2*t^5 + 1/2*t^4 - 4*t^3 - 5/2*t^2 + 7*t + 1/2,
 1/2*t^5 - 1/2*t^4 - 2*t^3 + 3/2*t^2 - 1/2,
 1/2*t^5 - 1/2*t^4 - 3*t^3 + 5/2*t^2 + 4*t - 5/2]
>>> CyclotomicField(Integer(12)).reduced_basis()
[1, zeta12^2, zeta12, zeta12^3]

reduced_gram_matrix(prec=None)[source]¶

Return the Gram matrix of an LLL-reduced basis for the Minkowski embedding of the maximal order of a number field.

INPUT:

prec – (default: None) the precision with which to calculate the Minkowski embedding (see NOTE below)

OUTPUT: the Gram matrix \([\langle x_i,x_j \rangle]\) of an LLL reduced basis for the maximal order of self, where the integral basis for self is given by \(\{x_0, \dots, x_{n-1}\}\). Here \(\langle , \rangle\) is the usual inner product on \(\RR^n\), and self is embedded in \(\RR^n\) by the Minkowski embedding. See the docstring for NumberField_absolute.minkowski_embedding() for more information.

Note

In the non-totally-real case, the LLL routine we call is currently PARI’s pari:qflll, which works with floating point approximations, and so the result is only as good as the precision promised by PARI. In particular, in this case, the returned matrix will not be integral, and may not have enough precision to recover the correct Gram matrix (which is known to be integral for theoretical reasons). Thus the need for the prec parameter above.

If the following run-time error occurs: “PariError: not a definite matrix in lllgram (42)”, try increasing the prec parameter,

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: F.<t> = NumberField(x^6 - 7*x^4 - x^3 + 11*x^2 + x - 1)
sage: F.reduced_gram_matrix()
[ 6  3  0  2  0  1]
[ 3  9  0  1  0 -2]
[ 0  0 14  6 -2  3]
[ 2  1  6 16 -3  3]
[ 0  0 -2 -3 16  6]
[ 1 -2  3  3  6 19]
sage: Matrix(6, [(x*y).trace()
....:            for x in F.integral_basis() for y in F.integral_basis()])
[2550  133  259  664 1368 3421]
[ 133   14    3   54   30  233]
[ 259    3   54   30  233  217]
[ 664   54   30  233  217 1078]
[1368   30  233  217 1078 1371]
[3421  233  217 1078 1371 5224]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> F = NumberField(x**Integer(6) - Integer(7)*x**Integer(4) - x**Integer(3) + Integer(11)*x**Integer(2) + x - Integer(1), names=('t',)); (t,) = F._first_ngens(1)
>>> F.reduced_gram_matrix()
[ 6  3  0  2  0  1]
[ 3  9  0  1  0 -2]
[ 0  0 14  6 -2  3]
[ 2  1  6 16 -3  3]
[ 0  0 -2 -3 16  6]
[ 1 -2  3  3  6 19]
>>> Matrix(Integer(6), [(x*y).trace()
...            for x in F.integral_basis() for y in F.integral_basis()])
[2550  133  259  664 1368 3421]
[ 133   14    3   54   30  233]
[ 259    3   54   30  233  217]
[ 664   54   30  233  217 1078]
[1368   30  233  217 1078 1371]
[3421  233  217 1078 1371 5224]

Sage

sage: x = polygen(QQ)
sage: F.<alpha> = NumberField(x^4 + x^2 + 712312*x + 131001238)
sage: F.reduced_gram_matrix(prec=128)
[   4.0000000000000000000000000000000000000   0.00000000000000000000000000000000000000   -1.9999999999999999999999999999999999037  -0.99999999999999999999999999999999383702]
[  0.00000000000000000000000000000000000000    46721.539331563218381658483353092335550   -11488.910026551724275122749703614966768   -418.12718083977141198754424579680468382]
[  -1.9999999999999999999999999999999999037   -11488.910026551724275122749703614966768  5.5658915310500611768713076521847709187e8  1.4179092271494070050433368847682152174e8]
[ -0.99999999999999999999999999999999383702   -418.12718083977141198754424579680468382  1.4179092271494070050433368847682152174e8 1.3665897267919181137884111201405279175e12]

Python

>>> from sage.all import *
>>> x = polygen(QQ)
>>> F = NumberField(x**Integer(4) + x**Integer(2) + Integer(712312)*x + Integer(131001238), names=('alpha',)); (alpha,) = F._first_ngens(1)
>>> F.reduced_gram_matrix(prec=Integer(128))
[   4.0000000000000000000000000000000000000   0.00000000000000000000000000000000000000   -1.9999999999999999999999999999999999037  -0.99999999999999999999999999999999383702]
[  0.00000000000000000000000000000000000000    46721.539331563218381658483353092335550   -11488.910026551724275122749703614966768   -418.12718083977141198754424579680468382]
[  -1.9999999999999999999999999999999999037   -11488.910026551724275122749703614966768  5.5658915310500611768713076521847709187e8  1.4179092271494070050433368847682152174e8]
[ -0.99999999999999999999999999999999383702   -418.12718083977141198754424579680468382  1.4179092271494070050433368847682152174e8 1.3665897267919181137884111201405279175e12]

regulator(proof=None)[source]¶

Return the regulator of this number field.

Note that PARI computes the regulator to higher precision than the Sage default.

INPUT:

proof – (default: True) unless you set it otherwise

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 - 2, 'a').regulator()
0.881373587019543
sage: NumberField(x^4 + x^3 + x^2 + x + 1, 'a').regulator()
0.962423650119207

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) - Integer(2), 'a').regulator()
0.881373587019543
>>> NumberField(x**Integer(4) + x**Integer(3) + x**Integer(2) + x + Integer(1), 'a').regulator()
0.962423650119207

residue_field(prime, names=None, check=True)[source]¶

Return the residue field of this number field at a given prime, ie \(O_K / p O_K\).

INPUT:

prime – a prime ideal of the maximal order in this number field, or an element of the field which generates a principal prime ideal.
names – the name of the variable in the residue field
check – whether or not to check the primality of prime

OUTPUT: the residue field at this prime

EXAMPLES:

Sage

sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^4 + 3*x^2 - 17)
sage: P = K.ideal(61).factor()[0][0]
sage: K.residue_field(P)
Residue field in abar of Fractional ideal (61, a^2 + 30)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(4) + Integer(3)*x**Integer(2) - Integer(17), names=('a',)); (a,) = K._first_ngens(1)
>>> P = K.ideal(Integer(61)).factor()[Integer(0)][Integer(0)]
>>> K.residue_field(P)
Residue field in abar of Fractional ideal (61, a^2 + 30)

Sage

sage: K.<i> = NumberField(x^2 + 1)
sage: K.residue_field(1+i)
Residue field of Fractional ideal (i + 1)

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.residue_field(Integer(1)+i)
Residue field of Fractional ideal (i + 1)

roots_of_unity()[source]¶

Return all the roots of unity in this field, primitive or not.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<b> = NumberField(x^2 + 1)
sage: zs = K.roots_of_unity(); zs
[b, -1, -b, 1]
sage: [z**K.number_of_roots_of_unity() for z in zs]
[1, 1, 1, 1]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('b',)); (b,) = K._first_ngens(1)
>>> zs = K.roots_of_unity(); zs
[b, -1, -b, 1]
>>> [z**K.number_of_roots_of_unity() for z in zs]
[1, 1, 1, 1]

selmer_generators(S, m, proof=True, orders=False)[source]¶

Compute generators of the group \(K(S,m)\).

INPUT:

S – set of primes of self
m – positive integer
proof – if False, assume the GRH in computing the class group
orders – boolean (default: False); if True, output two lists, the generators and their orders

OUTPUT:

A list of generators of \(K(S,m)\), and (optionally) their orders as elements of \(K^\times/(K^\times)^m\). This is the subgroup of \(K^\times/(K^\times)^m\) consisting of elements \(a\) such that the valuation of \(a\) is divisible by \(m\) at all primes not in \(S\). It fits in an exact sequence between the units modulo \(m\)-th powers and the \(m\)-torsion in the \(S\)-class group:

\[1 \longrightarrow O_{K,S}^\times / (O_{K,S}^\times)^m \longrightarrow K(S,m) \longrightarrow \operatorname{Cl}_{K,S}[m] \longrightarrow 0.\]

The group \(K(S,m)\) contains the subgroup of those \(a\) such that \(K(\sqrt[m]{a})/K\) is unramified at all primes of \(K\) outside of \(S\), but may contain it properly when not all primes dividing \(m\) are in \(S\).

See also

NumberField_generic.selmer_space(), which gives additional output when \(m=p\) is prime: as well as generators, it gives an abstract vector space over \(\GF{p}\) isomorphic to \(K(S,p)\) and maps implementing the isomorphism between this space and \(K(S,p)\) as a subgroup of \(K^*/(K^*)^p\).

EXAMPLES:

Sage

sage: K.<a> = QuadraticField(-5)
sage: K.selmer_generators((), 2)
[-1, 2]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> K.selmer_generators((), Integer(2))
[-1, 2]

The previous example shows that the group generated by the output may be strictly larger than the group of elements giving extensions unramified outside \(S\), since that has order just 2, generated by \(-1\):

Sage

sage: K.class_number()
2
sage: K.hilbert_class_field('b')
Number Field in b with defining polynomial x^2 + 1 over its base field

Python

>>> from sage.all import *
>>> K.class_number()
2
>>> K.hilbert_class_field('b')
Number Field in b with defining polynomial x^2 + 1 over its base field

When \(m\) is prime all the orders are equal to \(m\), but in general they are only divisors of \(m\):

Sage

sage: K.<a> = QuadraticField(-5)
sage: P2 = K.ideal(2, -a + 1)
sage: P3 = K.ideal(3, a + 1)
sage: K.selmer_generators((), 2, orders=True)
([-1, 2], [2, 2])
sage: K.selmer_generators((), 4, orders=True)
([-1, 4], [2, 2])
sage: K.selmer_generators([P2], 2)
[2, -1]
sage: K.selmer_generators((P2,P3), 4)
[2, -a - 1, -1]
sage: K.selmer_generators((P2,P3), 4, orders=True)
([2, -a - 1, -1], [4, 4, 2])
sage: K.selmer_generators([P2], 3)
[2]
sage: K.selmer_generators([P2, P3], 3)
[2, -a - 1]
sage: K.selmer_generators([P2, P3, K.ideal(a)], 3)  # random signs
[2, a + 1, a]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> P2 = K.ideal(Integer(2), -a + Integer(1))
>>> P3 = K.ideal(Integer(3), a + Integer(1))
>>> K.selmer_generators((), Integer(2), orders=True)
([-1, 2], [2, 2])
>>> K.selmer_generators((), Integer(4), orders=True)
([-1, 4], [2, 2])
>>> K.selmer_generators([P2], Integer(2))
[2, -1]
>>> K.selmer_generators((P2,P3), Integer(4))
[2, -a - 1, -1]
>>> K.selmer_generators((P2,P3), Integer(4), orders=True)
([2, -a - 1, -1], [4, 4, 2])
>>> K.selmer_generators([P2], Integer(3))
[2]
>>> K.selmer_generators([P2, P3], Integer(3))
[2, -a - 1]
>>> K.selmer_generators([P2, P3, K.ideal(a)], Integer(3))  # random signs
[2, a + 1, a]

Example over \(\QQ\) (as a number field):

Sage

sage: K.<a> = NumberField(polygen(QQ))
sage: K.selmer_generators([],5)
[]
sage: K.selmer_generators([K.prime_above(p) for p in [2,3,5]],2)
[2, 3, 5, -1]
sage: K.selmer_generators([K.prime_above(p) for p in [2,3,5]],6, orders=True)
([2, 3, 5, -1], [6, 6, 6, 2])

Python

>>> from sage.all import *
>>> K = NumberField(polygen(QQ), names=('a',)); (a,) = K._first_ngens(1)
>>> K.selmer_generators([],Integer(5))
[]
>>> K.selmer_generators([K.prime_above(p) for p in [Integer(2),Integer(3),Integer(5)]],Integer(2))
[2, 3, 5, -1]
>>> K.selmer_generators([K.prime_above(p) for p in [Integer(2),Integer(3),Integer(5)]],Integer(6), orders=True)
([2, 3, 5, -1], [6, 6, 6, 2])

selmer_group_iterator(S, m, proof=True)[source]¶

Return an iterator through elements of the finite group \(K(S,m)\).

INPUT:

S – set of primes of self
m – positive integer
proof – if False, assume the GRH in computing the class group

OUTPUT:

An iterator yielding the distinct elements of \(K(S,m)\). See the docstring for NumberField_generic.selmer_generators() for more information.

EXAMPLES:

Sage

sage: K.<a> = QuadraticField(-5)
sage: list(K.selmer_group_iterator((), 2))
[1, 2, -1, -2]
sage: list(K.selmer_group_iterator((), 4))
[1, 4, -1, -4]
sage: list(K.selmer_group_iterator([K.ideal(2, -a + 1)], 2))
[1, -1, 2, -2]
sage: list(K.selmer_group_iterator([K.ideal(2, -a + 1), K.ideal(3, a + 1)], 2))
[1, -1, -a - 1, a + 1, 2, -2, -2*a - 2, 2*a + 2]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> list(K.selmer_group_iterator((), Integer(2)))
[1, 2, -1, -2]
>>> list(K.selmer_group_iterator((), Integer(4)))
[1, 4, -1, -4]
>>> list(K.selmer_group_iterator([K.ideal(Integer(2), -a + Integer(1))], Integer(2)))
[1, -1, 2, -2]
>>> list(K.selmer_group_iterator([K.ideal(Integer(2), -a + Integer(1)), K.ideal(Integer(3), a + Integer(1))], Integer(2)))
[1, -1, -a - 1, a + 1, 2, -2, -2*a - 2, 2*a + 2]

Examples over \(\QQ\) (as a number field):

Sage

sage: K.<a> = NumberField(polygen(QQ))
sage: list(K.selmer_group_iterator([], 5))
[1]
sage: list(K.selmer_group_iterator([], 4))
[1, -1]
sage: list(K.selmer_group_iterator([K.prime_above(p) for p in [11,13]],2))
[1, -1, 13, -13, 11, -11, 143, -143]

Python

>>> from sage.all import *
>>> K = NumberField(polygen(QQ), names=('a',)); (a,) = K._first_ngens(1)
>>> list(K.selmer_group_iterator([], Integer(5)))
[1]
>>> list(K.selmer_group_iterator([], Integer(4)))
[1, -1]
>>> list(K.selmer_group_iterator([K.prime_above(p) for p in [Integer(11),Integer(13)]],Integer(2)))
[1, -1, 13, -13, 11, -11, 143, -143]

selmer_space(S, p, proof=None)[source]¶

Compute the group \(K(S,p)\) as a vector space with maps to and from \(K^*\).

INPUT:

S – set of primes ideals of self
p – a prime number
proof – if False, assume the GRH in computing the class group

OUTPUT:

(tuple) KSp, KSp_gens, from_KSp, to_KSp where

KSp is an abstract vector space over \(GF(p)\) isomorphic to \(K(S,p)\);
KSp_gens is a list of elements of \(K^*\) generating \(K(S,p)\);
from_KSp is a function from KSp to \(K^*\) implementing the isomorphism from the abstract \(K(S,p)\) to \(K(S,p)\) as a subgroup of \(K^*/(K^*)^p\);
to_KSP is a partial function from \(K^*\) to KSp, defined on elements \(a\) whose image in \(K^*/(K^*)^p\) lies in \(K(S,p)\), mapping them via the inverse isomorphism to the abstract vector space KSp.

The group \(K(S,p)\) is the finite subgroup of \(K^*/(K^*)^p\) consisting of elements whose valuation at all primes not in \(S\) is a multiple of \(p\). It contains the subgroup of those \(a\in K^*\) such that \(K(\sqrt[p]{a})/K\) is unramified at all primes of \(K\) outside of \(S\), but may contain it properly when not all primes dividing \(p\) are in \(S\).

EXAMPLES:

A real quadratic field with class number 2, where the fundamental unit is a generator, and the class group provides another generator when \(p=2\):

Sage

sage: K.<a> = QuadraticField(-5)
sage: K.class_number()
2
sage: P2 = K.ideal(2, -a + 1)
sage: P3 = K.ideal(3, a + 1)
sage: P5 = K.ideal(a)
sage: KS2, gens, fromKS2, toKS2 = K.selmer_space([P2, P3, P5], 2)
sage: KS2
Vector space of dimension 4 over Finite Field of size 2
sage: gens
[a + 1, a, 2, -1]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> K.class_number()
2
>>> P2 = K.ideal(Integer(2), -a + Integer(1))
>>> P3 = K.ideal(Integer(3), a + Integer(1))
>>> P5 = K.ideal(a)
>>> KS2, gens, fromKS2, toKS2 = K.selmer_space([P2, P3, P5], Integer(2))
>>> KS2
Vector space of dimension 4 over Finite Field of size 2
>>> gens
[a + 1, a, 2, -1]

Each generator must have even valuation at primes not in \(S\):

Sage

sage: [K.ideal(g).factor() for g in gens]
[(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1)),
 Fractional ideal (a),
 (Fractional ideal (2, a + 1))^2,
 1]

sage: toKS2(10)
(0, 0, 1, 1)
sage: fromKS2([0,0,1,1])
-2
sage: K(10/(-2)).is_square()
True

sage: KS3, gens, fromKS3, toKS3 = K.selmer_space([P2, P3, P5], 3)
sage: KS3
Vector space of dimension 3 over Finite Field of size 3
sage: gens
[1/2, 1/4*a + 1/4, a]

Python

>>> from sage.all import *
>>> [K.ideal(g).factor() for g in gens]
[(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1)),
 Fractional ideal (a),
 (Fractional ideal (2, a + 1))^2,
 1]

>>> toKS2(Integer(10))
(0, 0, 1, 1)
>>> fromKS2([Integer(0),Integer(0),Integer(1),Integer(1)])
-2
>>> K(Integer(10)/(-Integer(2))).is_square()
True

>>> KS3, gens, fromKS3, toKS3 = K.selmer_space([P2, P3, P5], Integer(3))
>>> KS3
Vector space of dimension 3 over Finite Field of size 3
>>> gens
[1/2, 1/4*a + 1/4, a]

An example to show that the group \(K(S,2)\) may be strictly larger than the group of elements giving extensions unramified outside \(S\). In this case, with \(K\) of class number \(2\) and \(S\) empty, there is only one quadratic extension of \(K\) unramified outside \(S\), the Hilbert Class Field \(K(\sqrt{-1})\):

Sage

sage: K.<a> = QuadraticField(-5)
sage: KS2, gens, fromKS2, toKS2 = K.selmer_space([], 2)
sage: KS2
Vector space of dimension 2 over Finite Field of size 2
sage: gens
[2, -1]
sage: x = polygen(ZZ, 'x')
sage: for v in KS2:
....:     if not v:
....:         continue
....:     a = fromKS2(v)
....:     print((a, K.extension(x^2 - a, 'roota').relative_discriminant().factor()))
(2, (Fractional ideal (2, a + 1))^4)
(-1, 1)
(-2, (Fractional ideal (2, a + 1))^4)

sage: K.hilbert_class_field('b')
Number Field in b with defining polynomial x^2 + 1 over its base field

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> KS2, gens, fromKS2, toKS2 = K.selmer_space([], Integer(2))
>>> KS2
Vector space of dimension 2 over Finite Field of size 2
>>> gens
[2, -1]
>>> x = polygen(ZZ, 'x')
>>> for v in KS2:
...     if not v:
...         continue
...     a = fromKS2(v)
...     print((a, K.extension(x**Integer(2) - a, 'roota').relative_discriminant().factor()))
(2, (Fractional ideal (2, a + 1))^4)
(-1, 1)
(-2, (Fractional ideal (2, a + 1))^4)

>>> K.hilbert_class_field('b')
Number Field in b with defining polynomial x^2 + 1 over its base field

signature()[source]¶

Return \((r_1, r_2)\), where \(r_1\) and \(r_2\) are the number of real embeddings and pairs of complex embeddings of this field, respectively.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 + 1, 'a').signature()
(0, 1)
sage: NumberField(x^3 - 2, 'a').signature()
(1, 1)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) + Integer(1), 'a').signature()
(0, 1)
>>> NumberField(x**Integer(3) - Integer(2), 'a').signature()
(1, 1)

solve_CRT(reslist, Ilist, check=True)[source]¶

Solve a Chinese remainder problem over this number field.

INPUT:

reslist – list of residues, i.e. integral number field elements
Ilist – list of integral ideals, assumed pairwise coprime
check – boolean (default: True); if True, result is checked

OUTPUT:

An integral element \(x\) such that x - reslist[i] is in Ilist[i] for all \(i\).

Note

The current implementation requires the ideals to be pairwise coprime. A more general version would be possible.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 - 10)
sage: Ilist = [K.primes_above(p)[0] for p in prime_range(10)]
sage: b = K.solve_CRT([1,2,3,4], Ilist, True)
sage: all(b - i - 1 in Ilist[i] for i in range(4))
True
sage: Ilist = [K.ideal(a), K.ideal(2)]
sage: K.solve_CRT([0,1], Ilist, True)
Traceback (most recent call last):
...
ArithmeticError: ideals in solve_CRT() must be pairwise coprime
sage: Ilist[0] + Ilist[1]
Fractional ideal (2, a)

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(10), names=('a',)); (a,) = K._first_ngens(1)
>>> Ilist = [K.primes_above(p)[Integer(0)] for p in prime_range(Integer(10))]
>>> b = K.solve_CRT([Integer(1),Integer(2),Integer(3),Integer(4)], Ilist, True)
>>> all(b - i - Integer(1) in Ilist[i] for i in range(Integer(4)))
True
>>> Ilist = [K.ideal(a), K.ideal(Integer(2))]
>>> K.solve_CRT([Integer(0),Integer(1)], Ilist, True)
Traceback (most recent call last):
...
ArithmeticError: ideals in solve_CRT() must be pairwise coprime
>>> Ilist[Integer(0)] + Ilist[Integer(1)]
Fractional ideal (2, a)

some_elements()[source]¶

Return a list of elements in the given number field.

EXAMPLES:

Sage

sage: R.<t> = QQ[]
sage: K.<a> =  QQ.extension(t^2 - 2); K
Number Field in a with defining polynomial t^2 - 2
sage: K.some_elements()
[1, a, 2*a, 3*a - 4, 1/2, 1/3*a, 1/6*a, 0, 1/2*a, 2, ..., 12, -12*a + 18]

sage: T.<u> = K[]
sage: M.<b> = K.extension(t^3 - 5); M
Number Field in b with defining polynomial t^3 - 5 over its base field
sage: M.some_elements()
[1, b, 1/2*a*b, ..., 2/5*b^2 + 2/5, 1/6*b^2 + 5/6*b + 13/6, 2]

Python

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> K = QQ.extension(t**Integer(2) - Integer(2), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial t^2 - 2
>>> K.some_elements()
[1, a, 2*a, 3*a - 4, 1/2, 1/3*a, 1/6*a, 0, 1/2*a, 2, ..., 12, -12*a + 18]

>>> T = K['u']; (u,) = T._first_ngens(1)
>>> M = K.extension(t**Integer(3) - Integer(5), names=('b',)); (b,) = M._first_ngens(1); M
Number Field in b with defining polynomial t^3 - 5 over its base field
>>> M.some_elements()
[1, b, 1/2*a*b, ..., 2/5*b^2 + 2/5, 1/6*b^2 + 5/6*b + 13/6, 2]

specified_complex_embedding()[source]¶

Return the embedding of this field into the complex numbers which has been specified.

Fields created with the QuadraticField() or CyclotomicField() constructors come with an implicit embedding. To get one of these fields without the embedding, use the generic NumberField constructor.

EXAMPLES:

Sage

sage: QuadraticField(-1, 'I').specified_complex_embedding()
Generic morphism:
  From: Number Field in I with defining polynomial x^2 + 1 with I = 1*I
  To:   Complex Lazy Field
  Defn: I -> 1*I

Python

>>> from sage.all import *
>>> QuadraticField(-Integer(1), 'I').specified_complex_embedding()
Generic morphism:
  From: Number Field in I with defining polynomial x^2 + 1 with I = 1*I
  To:   Complex Lazy Field
  Defn: I -> 1*I

Sage

sage: QuadraticField(3, 'a').specified_complex_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^2 - 3
        with a = 1.732050807568878?
  To:   Real Lazy Field
  Defn: a -> 1.732050807568878?

Python

>>> from sage.all import *
>>> QuadraticField(Integer(3), 'a').specified_complex_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^2 - 3
        with a = 1.732050807568878?
  To:   Real Lazy Field
  Defn: a -> 1.732050807568878?

Sage

sage: CyclotomicField(13).specified_complex_embedding()
Generic morphism:
  From: Cyclotomic Field of order 13 and degree 12
  To:   Complex Lazy Field
  Defn: zeta13 -> 0.885456025653210? + 0.464723172043769?*I

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(13)).specified_complex_embedding()
Generic morphism:
  From: Cyclotomic Field of order 13 and degree 12
  To:   Complex Lazy Field
  Defn: zeta13 -> 0.885456025653210? + 0.464723172043769?*I

Most fields don’t implicitly have embeddings unless explicitly specified:

Sage

sage: x = polygen(QQ, 'x')
sage: NumberField(x^2 - 2, 'a').specified_complex_embedding() is None
True
sage: NumberField(x^3 - x + 5, 'a').specified_complex_embedding() is None
True
sage: NumberField(x^3 - x + 5, 'a', embedding=2).specified_complex_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - x + 5
        with a = -1.904160859134921?
  To:   Real Lazy Field
  Defn: a -> -1.904160859134921?
sage: NumberField(x^3 - x + 5, 'a', embedding=CDF.0).specified_complex_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - x + 5
        with a = 0.952080429567461? + 1.311248044077123?*I
  To:   Complex Lazy Field
  Defn: a -> 0.952080429567461? + 1.311248044077123?*I

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> NumberField(x**Integer(2) - Integer(2), 'a').specified_complex_embedding() is None
True
>>> NumberField(x**Integer(3) - x + Integer(5), 'a').specified_complex_embedding() is None
True
>>> NumberField(x**Integer(3) - x + Integer(5), 'a', embedding=Integer(2)).specified_complex_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - x + 5
        with a = -1.904160859134921?
  To:   Real Lazy Field
  Defn: a -> -1.904160859134921?
>>> NumberField(x**Integer(3) - x + Integer(5), 'a', embedding=CDF.gen(0)).specified_complex_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^3 - x + 5
        with a = 0.952080429567461? + 1.311248044077123?*I
  To:   Complex Lazy Field
  Defn: a -> 0.952080429567461? + 1.311248044077123?*I

This function only returns complex embeddings:

Sage

sage: # needs sage.rings.padics
sage: K.<a> = NumberField(x^2 - 2, embedding=Qp(7)(2).sqrt())
sage: K.specified_complex_embedding() is None
True
sage: K.gen_embedding()
3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8 + 6*7^9 + 6*7^10
 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16 + 7^17 + 4*7^18 + 6*7^19 + O(7^20)
sage: K.coerce_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^2 - 2
        with a = 3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8
                  + 6*7^9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16
                  + 7^17 + 4*7^18 + 6*7^19 + O(7^20)
  To:   7-adic Field with capped relative precision 20
  Defn: a -> 3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8
              + 6*7^9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16
              + 7^17 + 4*7^18 + 6*7^19 + O(7^20)

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> K = NumberField(x**Integer(2) - Integer(2), embedding=Qp(Integer(7))(Integer(2)).sqrt(), names=('a',)); (a,) = K._first_ngens(1)
>>> K.specified_complex_embedding() is None
True
>>> K.gen_embedding()
3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8 + 6*7^9 + 6*7^10
 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16 + 7^17 + 4*7^18 + 6*7^19 + O(7^20)
>>> K.coerce_embedding()
Generic morphism:
  From: Number Field in a with defining polynomial x^2 - 2
        with a = 3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8
                  + 6*7^9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16
                  + 7^17 + 4*7^18 + 6*7^19 + O(7^20)
  To:   7-adic Field with capped relative precision 20
  Defn: a -> 3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8
              + 6*7^9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16
              + 7^17 + 4*7^18 + 6*7^19 + O(7^20)

structure()[source]¶

Return fixed isomorphism or embedding structure on self.

This is used to record various isomorphisms or embeddings that arise naturally in other constructions.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<z> = NumberField(x^2 + 3)
sage: L.<a> = K.absolute_field(); L
Number Field in a with defining polynomial x^2 + 3
sage: L.structure()
(Isomorphism given by variable name change map:
  From: Number Field in a with defining polynomial x^2 + 3
  To:   Number Field in z with defining polynomial x^2 + 3,
 Isomorphism given by variable name change map:
  From: Number Field in z with defining polynomial x^2 + 3
  To:   Number Field in a with defining polynomial x^2 + 3)

sage: K.<a> = QuadraticField(-3)
sage: R.<y> = K[]
sage: D.<x0> = K.extension(y)
sage: D_abs.<y0> = D.absolute_field()
sage: D_abs.structure()[0](y0)
-a

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> L = K.absolute_field(names=('a',)); (a,) = L._first_ngens(1); L
Number Field in a with defining polynomial x^2 + 3
>>> L.structure()
(Isomorphism given by variable name change map:
  From: Number Field in a with defining polynomial x^2 + 3
  To:   Number Field in z with defining polynomial x^2 + 3,
 Isomorphism given by variable name change map:
  From: Number Field in z with defining polynomial x^2 + 3
  To:   Number Field in a with defining polynomial x^2 + 3)

>>> K = QuadraticField(-Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> R = K['y']; (y,) = R._first_ngens(1)
>>> D = K.extension(y, names=('x0',)); (x0,) = D._first_ngens(1)
>>> D_abs = D.absolute_field(names=('y0',)); (y0,) = D_abs._first_ngens(1)
>>> D_abs.structure()[Integer(0)](y0)
-a

subfield(alpha, name=None, names=None)[source]¶

Return a number field \(K\) isomorphic to \(\QQ(\alpha)\) (if this is an absolute number field) or \(L(\alpha)\) (if this is a relative extension \(M/L\)) and a map from \(K\) to self that sends the generator of \(K\) to alpha.

INPUT:

alpha – an element of self, or something that coerces to an element of self

OUTPUT:

K – a number field
from_K – a homomorphism from \(K\) to self that sends the generator of \(K\) to alpha

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^4 - 3); K
Number Field in a with defining polynomial x^4 - 3
sage: H.<b>, from_H = K.subfield(a^2)
sage: H
Number Field in b with defining polynomial x^2 - 3 with b = a^2
sage: from_H(b)
a^2
sage: from_H
Ring morphism:
  From: Number Field in b with defining polynomial x^2 - 3 with b = a^2
  To:   Number Field in a with defining polynomial x^4 - 3
  Defn: b |--> a^2

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(4) - Integer(3), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^4 - 3
>>> H, from_H  = K.subfield(a**Integer(2), names=('b',)); (b,) = H._first_ngens(1)
>>> H
Number Field in b with defining polynomial x^2 - 3 with b = a^2
>>> from_H(b)
a^2
>>> from_H
Ring morphism:
  From: Number Field in b with defining polynomial x^2 - 3 with b = a^2
  To:   Number Field in a with defining polynomial x^4 - 3
  Defn: b |--> a^2

A relative example. Note that the result returned is the subfield generated by \(\alpha\) over self.base_field(), not over \(\QQ\) (see Issue #5392):

Sage

sage: L.<a> = NumberField(x^2 - 3)
sage: M.<b> = L.extension(x^4 + 1)
sage: K, phi = M.subfield(b^2)
sage: K.base_field() is L
True

Python

>>> from sage.all import *
>>> L = NumberField(x**Integer(2) - Integer(3), names=('a',)); (a,) = L._first_ngens(1)
>>> M = L.extension(x**Integer(4) + Integer(1), names=('b',)); (b,) = M._first_ngens(1)
>>> K, phi = M.subfield(b**Integer(2))
>>> K.base_field() is L
True

Subfields inherit embeddings:

Sage

sage: K.<z> = CyclotomicField(5)
sage: L, K_from_L = K.subfield(z - z^2 - z^3 + z^4)
sage: L
Number Field in z0 with defining polynomial x^2 - 5 with z0 = 2.236067977499790?
sage: CLF_from_K = K.coerce_embedding(); CLF_from_K
Generic morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Lazy Field
  Defn: z -> 0.309016994374948? + 0.951056516295154?*I
sage: CLF_from_L = L.coerce_embedding(); CLF_from_L
Generic morphism:
  From: Number Field in z0 with defining polynomial x^2 - 5
        with z0 = 2.236067977499790?
  To:   Complex Lazy Field
  Defn: z0 -> 2.236067977499790?

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(5), names=('z',)); (z,) = K._first_ngens(1)
>>> L, K_from_L = K.subfield(z - z**Integer(2) - z**Integer(3) + z**Integer(4))
>>> L
Number Field in z0 with defining polynomial x^2 - 5 with z0 = 2.236067977499790?
>>> CLF_from_K = K.coerce_embedding(); CLF_from_K
Generic morphism:
  From: Cyclotomic Field of order 5 and degree 4
  To:   Complex Lazy Field
  Defn: z -> 0.309016994374948? + 0.951056516295154?*I
>>> CLF_from_L = L.coerce_embedding(); CLF_from_L
Generic morphism:
  From: Number Field in z0 with defining polynomial x^2 - 5
        with z0 = 2.236067977499790?
  To:   Complex Lazy Field
  Defn: z0 -> 2.236067977499790?

Check transitivity:

Sage

sage: CLF_from_L(L.gen())
2.236067977499790?
sage: CLF_from_K(K_from_L(L.gen()))
2.23606797749979? + 0.?e-14*I

Python

>>> from sage.all import *
>>> CLF_from_L(L.gen())
2.236067977499790?
>>> CLF_from_K(K_from_L(L.gen()))
2.23606797749979? + 0.?e-14*I

If self has no specified embedding, then \(K\) comes with an embedding in self:

Sage

sage: K.<a> = NumberField(x^6 - 6*x^4 + 8*x^2 - 1)
sage: L.<b>, from_L = K.subfield(a^2)
sage: L
Number Field in b with defining polynomial x^3 - 6*x^2 + 8*x - 1 with b = a^2
sage: L.gen_embedding()
a^2

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(6) - Integer(6)*x**Integer(4) + Integer(8)*x**Integer(2) - Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> L, from_L  = K.subfield(a**Integer(2), names=('b',)); (b,) = L._first_ngens(1)
>>> L
Number Field in b with defining polynomial x^3 - 6*x^2 + 8*x - 1 with b = a^2
>>> L.gen_embedding()
a^2

You can also view a number field as having a different generator by just choosing the input to generate the whole field; for that it is better to use change_generator(), which gives isomorphisms in both directions.

subfield_from_elements(alpha, name=None, polred=True, threshold=None)[source]¶

Return the subfield generated by the elements alpha.

If the generated subfield by the elements alpha is either the rational field or the complete number field, the field returned is respectively QQ or self.

INPUT:

alpha – list of elements in this number field
name – a name for the generator of the new number field
polred – boolean (default: True); whether to optimize the generator of the newly created field
threshold – positive number (default: None) threshold to be passed to the do_polred function

OUTPUT: a triple (field, beta, hom) where

field – a subfield of this number field
beta – list of elements of field corresponding to alpha
hom – inclusion homomorphism from field to self

EXAMPLES:

Sage

sage: x = polygen(QQ)
sage: poly = x^4 - 4*x^2 + 1
sage: emb = AA.polynomial_root(poly, RIF(0.51, 0.52))
sage: K.<a> = NumberField(poly, embedding=emb)
sage: sqrt2 = -a^3 + 3*a
sage: sqrt3 = -a^2 + 2
sage: assert sqrt2 ** 2 == 2 and sqrt3 ** 2 == 3
sage: L, elts, phi = K.subfield_from_elements([sqrt2, 1 - sqrt2/3])
sage: L
Number Field in a0 with defining polynomial x^2 - 2 with a0 = 1.414213562373095?
sage: elts
[a0, -1/3*a0 + 1]
sage: phi
Ring morphism:
  From: Number Field in a0 with defining polynomial x^2 - 2
        with a0 = 1.414213562373095?
  To:   Number Field in a with defining polynomial x^4 - 4*x^2 + 1
        with a = 0.5176380902050415?
  Defn: a0 |--> -a^3 + 3*a
sage: assert phi(elts[0]) == sqrt2
sage: assert phi(elts[1]) == 1 - sqrt2/3


sage: L, elts, phi = K.subfield_from_elements([1, sqrt3])
sage: assert phi(elts[0]) == 1
sage: assert phi(elts[1]) == sqrt3

sage: L, elts, phi = K.subfield_from_elements([sqrt2, sqrt3])
sage: phi
Identity endomorphism of Number Field in a with defining polynomial
 x^4 - 4*x^2 + 1 with a = 0.5176380902050415?

Python

>>> from sage.all import *
>>> x = polygen(QQ)
>>> poly = x**Integer(4) - Integer(4)*x**Integer(2) + Integer(1)
>>> emb = AA.polynomial_root(poly, RIF(RealNumber('0.51'), RealNumber('0.52')))
>>> K = NumberField(poly, embedding=emb, names=('a',)); (a,) = K._first_ngens(1)
>>> sqrt2 = -a**Integer(3) + Integer(3)*a
>>> sqrt3 = -a**Integer(2) + Integer(2)
>>> assert sqrt2 ** Integer(2) == Integer(2) and sqrt3 ** Integer(2) == Integer(3)
>>> L, elts, phi = K.subfield_from_elements([sqrt2, Integer(1) - sqrt2/Integer(3)])
>>> L
Number Field in a0 with defining polynomial x^2 - 2 with a0 = 1.414213562373095?
>>> elts
[a0, -1/3*a0 + 1]
>>> phi
Ring morphism:
  From: Number Field in a0 with defining polynomial x^2 - 2
        with a0 = 1.414213562373095?
  To:   Number Field in a with defining polynomial x^4 - 4*x^2 + 1
        with a = 0.5176380902050415?
  Defn: a0 |--> -a^3 + 3*a
>>> assert phi(elts[Integer(0)]) == sqrt2
>>> assert phi(elts[Integer(1)]) == Integer(1) - sqrt2/Integer(3)


>>> L, elts, phi = K.subfield_from_elements([Integer(1), sqrt3])
>>> assert phi(elts[Integer(0)]) == Integer(1)
>>> assert phi(elts[Integer(1)]) == sqrt3

>>> L, elts, phi = K.subfield_from_elements([sqrt2, sqrt3])
>>> phi
Identity endomorphism of Number Field in a with defining polynomial
 x^4 - 4*x^2 + 1 with a = 0.5176380902050415?

trace_dual_basis(b)[source]¶

Compute the dual basis of a basis of self with respect to the trace pairing.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 + x + 1)
sage: b = [1, 2*a, 3*a^2]
sage: T = K.trace_dual_basis(b); T
[4/31*a^2 - 6/31*a + 13/31, -9/62*a^2 - 1/31*a - 3/31, 2/31*a^2 - 3/31*a + 4/93]
sage: [(b[i]*T[j]).trace() for i in range(3) for j in range(3)]
[1, 0, 0, 0, 1, 0, 0, 0, 1]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> b = [Integer(1), Integer(2)*a, Integer(3)*a**Integer(2)]
>>> T = K.trace_dual_basis(b); T
[4/31*a^2 - 6/31*a + 13/31, -9/62*a^2 - 1/31*a - 3/31, 2/31*a^2 - 3/31*a + 4/93]
>>> [(b[i]*T[j]).trace() for i in range(Integer(3)) for j in range(Integer(3))]
[1, 0, 0, 0, 1, 0, 0, 0, 1]

trace_pairing(v)[source]¶

Return the matrix of the trace pairing on the elements of the list v.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<zeta3> = NumberField(x^2 + 3)
sage: K.trace_pairing([1, zeta3])
[ 2  0]
[ 0 -6]

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(3), names=('zeta3',)); (zeta3,) = K._first_ngens(1)
>>> K.trace_pairing([Integer(1), zeta3])
[ 2  0]
[ 0 -6]

uniformizer(P, others='positive')[source]¶

Return an element of self with valuation 1 at the prime ideal \(P\).

INPUT:

self – a number field
P – a prime ideal of self
others – either 'positive' (default), in which case the element will have nonnegative valuation at all other primes of self, or 'negative', in which case the element will have nonpositive valuation at all other primes of self

Note

When \(P\) is principal (e.g., always when self has class number one) the result may or may not be a generator of \(P\)!

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 + 5); K
Number Field in a with defining polynomial x^2 + 5
sage: P, Q = K.ideal(3).prime_factors()
sage: P
Fractional ideal (3, a + 1)
sage: pi = K.uniformizer(P); pi
a + 1
sage: K.ideal(pi).factor()
(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1))
sage: pi = K.uniformizer(P,'negative'); pi
1/2*a + 1/2
sage: K.ideal(pi).factor()
(Fractional ideal (2, a + 1))^-1 * (Fractional ideal (3, a + 1))

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial x^2 + 5
>>> P, Q = K.ideal(Integer(3)).prime_factors()
>>> P
Fractional ideal (3, a + 1)
>>> pi = K.uniformizer(P); pi
a + 1
>>> K.ideal(pi).factor()
(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1))
>>> pi = K.uniformizer(P,'negative'); pi
1/2*a + 1/2
>>> K.ideal(pi).factor()
(Fractional ideal (2, a + 1))^-1 * (Fractional ideal (3, a + 1))

Sage

sage: K = CyclotomicField(9)
sage: Plist = K.ideal(17).prime_factors()
sage: pilist = [K.uniformizer(P) for P in Plist]
sage: [pi.is_integral() for pi in pilist]
[True, True, True]
sage: [pi.valuation(P) for pi, P in zip(pilist, Plist)]
[1, 1, 1]
sage: [ pilist[i] in Plist[i] for i in range(len(Plist)) ]
[True, True, True]

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(9))
>>> Plist = K.ideal(Integer(17)).prime_factors()
>>> pilist = [K.uniformizer(P) for P in Plist]
>>> [pi.is_integral() for pi in pilist]
[True, True, True]
>>> [pi.valuation(P) for pi, P in zip(pilist, Plist)]
[1, 1, 1]
>>> [ pilist[i] in Plist[i] for i in range(len(Plist)) ]
[True, True, True]

Sage

sage: K.<t> = NumberField(x^4 - x^3 - 3*x^2 - x + 1)
sage: [K.uniformizer(P) for P,e in factor(K.ideal(2))]
[2]
sage: [K.uniformizer(P) for P,e in factor(K.ideal(3))]
[t - 1]
sage: [K.uniformizer(P) for P,e in factor(K.ideal(5))]
[t^2 - t + 1, t + 2, t - 2]
sage: [K.uniformizer(P) for P,e in factor(K.ideal(7))]  # representation varies, not tested
[t^2 + 3*t + 1]
sage: [K.uniformizer(P) for P,e in factor(K.ideal(67))]
[t + 23, t + 26, t - 32, t - 18]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(4) - x**Integer(3) - Integer(3)*x**Integer(2) - x + Integer(1), names=('t',)); (t,) = K._first_ngens(1)
>>> [K.uniformizer(P) for P,e in factor(K.ideal(Integer(2)))]
[2]
>>> [K.uniformizer(P) for P,e in factor(K.ideal(Integer(3)))]
[t - 1]
>>> [K.uniformizer(P) for P,e in factor(K.ideal(Integer(5)))]
[t^2 - t + 1, t + 2, t - 2]
>>> [K.uniformizer(P) for P,e in factor(K.ideal(Integer(7)))]  # representation varies, not tested
[t^2 + 3*t + 1]
>>> [K.uniformizer(P) for P,e in factor(K.ideal(Integer(67)))]
[t + 23, t + 26, t - 32, t - 18]

ALGORITHM:

Use PARI. More precisely, use the second component of pari:idealprimedec in the “positive” case. Use pari:idealappr with exponent of \(-1\) and invert the result in the “negative” case.

unit_group(proof=None)[source]¶

Return the unit group (including torsion) of this number field.

ALGORITHM: Uses PARI’s pari:bnfinit command.

INPUT:

proof – boolean (default: True); flag passed to PARI

Note

The group is cached.

See also

units() S_unit_group() S_units()

EXAMPLES:

Sage

sage: x = QQ['x'].0
sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3
sage: K = NumberField(A, 'a')
sage: U = K.unit_group(); U
Unit group with structure C10 x Z of Number Field in a
 with defining polynomial x^4 - 10*x^3 + 100*x^2 - 375*x + 1375
sage: U.gens()
(u0, u1)
sage: U.gens_values()  # random
[-1/275*a^3 + 7/55*a^2 - 6/11*a + 4, 1/275*a^3 + 4/55*a^2 - 5/11*a + 3]
sage: U.invariants()
(10, 0)
sage: [u.multiplicative_order() for u in U.gens()]
[10, +Infinity]

Python

>>> from sage.all import *
>>> x = QQ['x'].gen(0)
>>> A = x**Integer(4) - Integer(10)*x**Integer(3) + Integer(20)*Integer(5)*x**Integer(2) - Integer(15)*Integer(5)**Integer(2)*x + Integer(11)*Integer(5)**Integer(3)
>>> K = NumberField(A, 'a')
>>> U = K.unit_group(); U
Unit group with structure C10 x Z of Number Field in a
 with defining polynomial x^4 - 10*x^3 + 100*x^2 - 375*x + 1375
>>> U.gens()
(u0, u1)
>>> U.gens_values()  # random
[-1/275*a^3 + 7/55*a^2 - 6/11*a + 4, 1/275*a^3 + 4/55*a^2 - 5/11*a + 3]
>>> U.invariants()
(10, 0)
>>> [u.multiplicative_order() for u in U.gens()]
[10, +Infinity]

For big number fields, provably computing the unit group can take a very long time. In this case, one can ask for the conjectural unit group (correct if the Generalized Riemann Hypothesis is true):

Sage

sage: K = NumberField(x^17 + 3, 'a')
sage: K.unit_group(proof=True)  # takes forever, not tested
...
sage: U = K.unit_group(proof=False)
sage: U
Unit group with structure C2 x Z x Z x Z x Z x Z x Z x Z x Z of
 Number Field in a with defining polynomial x^17 + 3
sage: U.gens()
(u0, u1, u2, u3, u4, u5, u6, u7, u8)
sage: U.gens_values()  # result not independently verified
[-1,
 -a^9 - a + 1,
 -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1,
 2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4,
 a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1,
 -a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2,
 -2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4,
 a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
 2*a^16 + a^15 - a^11 - 3*a^10 - 4*a^9 - 4*a^8 - 4*a^7 - 5*a^6 - 7*a^5 - 8*a^4 - 6*a^3 - 5*a^2 - 6*a - 7]

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(17) + Integer(3), 'a')
>>> K.unit_group(proof=True)  # takes forever, not tested
...
>>> U = K.unit_group(proof=False)
>>> U
Unit group with structure C2 x Z x Z x Z x Z x Z x Z x Z x Z of
 Number Field in a with defining polynomial x^17 + 3
>>> U.gens()
(u0, u1, u2, u3, u4, u5, u6, u7, u8)
>>> U.gens_values()  # result not independently verified
[-1,
 -a^9 - a + 1,
 -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1,
 2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4,
 a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1,
 -a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2,
 -2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4,
 a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
 2*a^16 + a^15 - a^11 - 3*a^10 - 4*a^9 - 4*a^8 - 4*a^7 - 5*a^6 - 7*a^5 - 8*a^4 - 6*a^3 - 5*a^2 - 6*a - 7]

units(proof=None)[source]¶

Return generators for the unit group modulo torsion.

ALGORITHM: Uses PARI’s pari:bnfinit command.

INPUT:

proof – boolean (default: True); flag passed to PARI

Note

For more functionality see unit_group().

EXAMPLES:

Sage

sage: x = polygen(QQ)
sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3
sage: K = NumberField(A, 'a')
sage: K.units()
(-1/275*a^3 - 4/55*a^2 + 5/11*a - 3,)

Python

>>> from sage.all import *
>>> x = polygen(QQ)
>>> A = x**Integer(4) - Integer(10)*x**Integer(3) + Integer(20)*Integer(5)*x**Integer(2) - Integer(15)*Integer(5)**Integer(2)*x + Integer(11)*Integer(5)**Integer(3)
>>> K = NumberField(A, 'a')
>>> K.units()
(-1/275*a^3 - 4/55*a^2 + 5/11*a - 3,)

For big number fields, provably computing the unit group can take a very long time. In this case, one can ask for the conjectural unit group (correct if the Generalized Riemann Hypothesis is true):

Sage

sage: K = NumberField(x^17 + 3, 'a')
sage: K.units(proof=True)  # takes forever, not tested
...
sage: K.units(proof=False)  # result not independently verified
(-a^9 - a + 1,
 -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1,
 2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4,
 a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1,
 -a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2,
 -2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4,
 a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
 2*a^16 + a^15 - a^11 - 3*a^10 - 4*a^9 - 4*a^8 - 4*a^7 - 5*a^6 - 7*a^5 - 8*a^4 - 6*a^3 - 5*a^2 - 6*a - 7)

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(17) + Integer(3), 'a')
>>> K.units(proof=True)  # takes forever, not tested
...
>>> K.units(proof=False)  # result not independently verified
(-a^9 - a + 1,
 -a^16 + a^15 - a^14 + a^12 - a^11 + a^10 + a^8 - a^7 + 2*a^6 - a^4 + 3*a^3 - 2*a^2 + 2*a - 1,
 2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a + 4,
 a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1,
 -a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 - a^7 - a^6 - a^5 - a^4 - a^3 - a^2 + 2,
 -2*a^16 + 3*a^15 - 3*a^14 + 3*a^13 - 3*a^12 + a^11 - a^9 + 3*a^8 - 4*a^7 + 5*a^6 - 6*a^5 + 4*a^4 - 3*a^3 + 2*a^2 + 2*a - 4,
 a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2 - 2*a + 2,
 2*a^16 + a^15 - a^11 - 3*a^10 - 4*a^9 - 4*a^8 - 4*a^7 - 5*a^6 - 7*a^5 - 8*a^4 - 6*a^3 - 5*a^2 - 6*a - 7)

valuation(prime)[source]¶

Return the valuation on this field defined by prime.

INPUT:

prime – a prime that does not split, a discrete (pseudo-)valuation or a fractional ideal

EXAMPLES:

The valuation can be specified with an integer prime that is completely ramified in R:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 + 1)
sage: K.valuation(2)                                                        # needs sage.rings.padics
2-adic valuation

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> K.valuation(Integer(2))                                                        # needs sage.rings.padics
2-adic valuation

It can also be unramified in R:

Sage

sage: K.valuation(3)                                                        # needs sage.rings.padics
3-adic valuation

Python

>>> from sage.all import *
>>> K.valuation(Integer(3))                                                        # needs sage.rings.padics
3-adic valuation

A prime that factors into pairwise distinct factors, results in an error:

Sage

sage: K.valuation(5)                                                        # needs sage.rings.padics
Traceback (most recent call last):
...
ValueError: The valuation Gauss valuation induced by 5-adic valuation does not
approximate a unique extension of 5-adic valuation with respect to x^2 + 1

Python

>>> from sage.all import *
>>> K.valuation(Integer(5))                                                        # needs sage.rings.padics
Traceback (most recent call last):
...
ValueError: The valuation Gauss valuation induced by 5-adic valuation does not
approximate a unique extension of 5-adic valuation with respect to x^2 + 1

The valuation can also be selected by giving a valuation on the base ring that extends uniquely:

Sage

sage: CyclotomicField(5).valuation(ZZ.valuation(5))                         # needs sage.rings.padics
5-adic valuation

Python

>>> from sage.all import *
>>> CyclotomicField(Integer(5)).valuation(ZZ.valuation(Integer(5)))                         # needs sage.rings.padics
5-adic valuation

When the extension is not unique, this does not work:

Sage

sage: K.valuation(ZZ.valuation(5))                                          # needs sage.rings.padics
Traceback (most recent call last):
...
ValueError: The valuation Gauss valuation induced by 5-adic valuation does not
approximate a unique extension of 5-adic valuation with respect to x^2 + 1

Python

>>> from sage.all import *
>>> K.valuation(ZZ.valuation(Integer(5)))                                          # needs sage.rings.padics
Traceback (most recent call last):
...
ValueError: The valuation Gauss valuation induced by 5-adic valuation does not
approximate a unique extension of 5-adic valuation with respect to x^2 + 1

For a number field which is of the form \(K[x]/(G)\), you can specify a valuation by providing a discrete pseudo-valuation on \(K[x]\) which sends \(G\) to infinity. This lets us specify which extension of the 5-adic valuation we care about in the above example:

Sage

sage: # needs sage.rings.padics
sage: R.<x> = QQ[]
sage: G5 = GaussValuation(R, QQ.valuation(5))
sage: v = K.valuation(G5.augmentation(x + 2, infinity))
sage: w = K.valuation(G5.augmentation(x + 1/2, infinity))
sage: v == w
False

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> G5 = GaussValuation(R, QQ.valuation(Integer(5)))
>>> v = K.valuation(G5.augmentation(x + Integer(2), infinity))
>>> w = K.valuation(G5.augmentation(x + Integer(1)/Integer(2), infinity))
>>> v == w
False

Note that you get the same valuation, even if you write down the pseudo-valuation differently:

Sage

sage: # needs sage.rings.padics
sage: ww = K.valuation(G5.augmentation(x + 3, infinity))
sage: w is ww
True

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> ww = K.valuation(G5.augmentation(x + Integer(3), infinity))
>>> w is ww
True

The valuation prime does not need to send the defining polynomial \(G\) to infinity. It is sufficient if it singles out one of the valuations on the number field. This is important if the prime only factors over the completion, i.e., if it is not possible to write down one of the factors within the number field:

Sage

sage: # needs sage.rings.padics
sage: v = G5.augmentation(x + 3, 1)
sage: K.valuation(v)
[ 5-adic valuation, v(x + 3) = 1 ]-adic valuation

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> v = G5.augmentation(x + Integer(3), Integer(1))
>>> K.valuation(v)
[ 5-adic valuation, v(x + 3) = 1 ]-adic valuation

Finally, prime can also be a fractional ideal of a number field if it singles out an extension of a \(p\)-adic valuation of the base field:

Sage

sage: K.valuation(K.fractional_ideal(a + 1))                                # needs sage.rings.padics
2-adic valuation

Python

>>> from sage.all import *
>>> K.valuation(K.fractional_ideal(a + Integer(1)))                                # needs sage.rings.padics
2-adic valuation

zeta(n=2, all=False)[source]¶

Return one, or a list of all, primitive \(n\)-th root of unity in this field.

INPUT:

n – positive integer
all – boolean; if False (default), return a primitive \(n\)-th root of unity in this field, or raise a ValueError exception if there are none. If True, return a list of all primitive \(n\)-th roots of unity in this field (possibly empty).

Note

To obtain the maximal order of a root of unity in this field, use number_of_roots_of_unity().

Note

We do not create the full unit group since that can be expensive, but we do use it if it is already known.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<z> = NumberField(x^2 + 3)
sage: K.zeta(1)
1
sage: K.zeta(2)
-1
sage: K.zeta(2, all=True)
[-1]
sage: K.zeta(3)
-1/2*z - 1/2
sage: K.zeta(3, all=True)
[-1/2*z - 1/2, 1/2*z - 1/2]
sage: K.zeta(4)
Traceback (most recent call last):
...
ValueError: there are no 4th roots of unity in self

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(3), names=('z',)); (z,) = K._first_ngens(1)
>>> K.zeta(Integer(1))
1
>>> K.zeta(Integer(2))
-1
>>> K.zeta(Integer(2), all=True)
[-1]
>>> K.zeta(Integer(3))
-1/2*z - 1/2
>>> K.zeta(Integer(3), all=True)
[-1/2*z - 1/2, 1/2*z - 1/2]
>>> K.zeta(Integer(4))
Traceback (most recent call last):
...
ValueError: there are no 4th roots of unity in self

Sage

sage: r.<x> = QQ[]
sage: K.<b> = NumberField(x^2 + 1)
sage: K.zeta(4)
b
sage: K.zeta(4,all=True)
[b, -b]
sage: K.zeta(3)
Traceback (most recent call last):
...
ValueError: there are no 3rd roots of unity in self
sage: K.zeta(3, all=True)
[]

Python

>>> from sage.all import *
>>> r = QQ['x']; (x,) = r._first_ngens(1)
>>> K = NumberField(x**Integer(2) + Integer(1), names=('b',)); (b,) = K._first_ngens(1)
>>> K.zeta(Integer(4))
b
>>> K.zeta(Integer(4),all=True)
[b, -b]
>>> K.zeta(Integer(3))
Traceback (most recent call last):
...
ValueError: there are no 3rd roots of unity in self
>>> K.zeta(Integer(3), all=True)
[]

Number fields defined by non-monic and non-integral polynomials are supported (Issue #252):

Sage

sage: K.<a> = NumberField(1/2*x^2 + 1/6)
sage: K.zeta(3)
-3/2*a - 1/2

Python

>>> from sage.all import *
>>> K = NumberField(Integer(1)/Integer(2)*x**Integer(2) + Integer(1)/Integer(6), names=('a',)); (a,) = K._first_ngens(1)
>>> K.zeta(Integer(3))
-3/2*a - 1/2

zeta_coefficients(n)[source]¶

Compute the first \(n\) coefficients of the Dedekind zeta function of this field as a Dirichlet series.

EXAMPLES:

Sage

sage: x = QQ['x'].0
sage: NumberField(x^2 + 1, 'a').zeta_coefficients(10)
[1, 1, 0, 1, 2, 0, 0, 1, 1, 2]

Python

>>> from sage.all import *
>>> x = QQ['x'].gen(0)
>>> NumberField(x**Integer(2) + Integer(1), 'a').zeta_coefficients(Integer(10))
[1, 1, 0, 1, 2, 0, 0, 1, 1, 2]

zeta_order()[source]¶

Return the number of roots of unity in this field.

Note

We do not create the full unit group since that can be expensive, but we do use it if it is already known.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: F.<alpha> = NumberField(x^22 + 3)
sage: F.zeta_order()
6
sage: F.<alpha> = NumberField(x^2 - 7)
sage: F.zeta_order()
2

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> F = NumberField(x**Integer(22) + Integer(3), names=('alpha',)); (alpha,) = F._first_ngens(1)
>>> F.zeta_order()
6
>>> F = NumberField(x**Integer(2) - Integer(7), names=('alpha',)); (alpha,) = F._first_ngens(1)
>>> F.zeta_order()
2

sage.rings.number_field.number_field.NumberField_generic_v1(poly, name, latex_name, canonical_embedding=None)[source]¶

Used for unpickling old pickles.

EXAMPLES:

Sage

sage: from sage.rings.number_field.number_field import NumberField_absolute_v1
sage: R.<x> = QQ[]
sage: NumberField_absolute_v1(x^2 + 1, 'i', 'i')
Number Field in i with defining polynomial x^2 + 1

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field import NumberField_absolute_v1
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> NumberField_absolute_v1(x**Integer(2) + Integer(1), 'i', 'i')
Number Field in i with defining polynomial x^2 + 1

class sage.rings.number_field.number_field.NumberField_quadratic(polynomial, name=None, latex_name=None, check=True, embedding=None, assume_disc_small=False, maximize_at_primes=None, structure=None)[source]¶

Bases: NumberField_absolute, NumberField_quadratic

Create a quadratic extension of the rational field.

The command QuadraticField(a) creates the field \(\QQ(\sqrt{a})\).

EXAMPLES:

Sage

sage: QuadraticField(3, 'a')
Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878?
sage: QuadraticField(-4, 'b')
Number Field in b with defining polynomial x^2 + 4 with b = 2*I

Python

>>> from sage.all import *
>>> QuadraticField(Integer(3), 'a')
Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878?
>>> QuadraticField(-Integer(4), 'b')
Number Field in b with defining polynomial x^2 + 4 with b = 2*I

class_number(proof=None)[source]¶

Return the size of the class group of self.

INPUT:

proof – boolean (default: True, unless you called proof.number_field() and set it otherwise). If proof is False (not the default!), and the discriminant of the field is negative, then the following warning from the PARI manual applies:

Warning

For \(D<0\), this function may give incorrect results when the class group has a low exponent (has many cyclic factors), because implementing Shank’s method in full generality slows it down immensely.

EXAMPLES:

Sage

sage: QuadraticField(-23,'a').class_number()
3

Python

>>> from sage.all import *
>>> QuadraticField(-Integer(23),'a').class_number()
3

These are all the primes so that the class number of \(\QQ(\sqrt{-p})\) is \(1\):

Sage

sage: [d for d in prime_range(2,300)
....:  if not is_square(d) and QuadraticField(-d,'a').class_number() == 1]
[2, 3, 7, 11, 19, 43, 67, 163]

Python

>>> from sage.all import *
>>> [d for d in prime_range(Integer(2),Integer(300))
...  if not is_square(d) and QuadraticField(-d,'a').class_number() == Integer(1)]
[2, 3, 7, 11, 19, 43, 67, 163]

It is an open problem to prove that there are infinity many positive square-free \(d\) such that \(\QQ(\sqrt{d})\) has class number \(1\):

Sage

sage: len([d for d in range(2,200)
....:      if not is_square(d) and QuadraticField(d,'a').class_number() == 1])
121

Python

>>> from sage.all import *
>>> len([d for d in range(Integer(2),Integer(200))
...      if not is_square(d) and QuadraticField(d,'a').class_number() == Integer(1)])
121

discriminant(v=None)[source]¶

Return the discriminant of the ring of integers of the number field, or if v is specified, the determinant of the trace pairing on the elements of the list v.

INPUT:

v – (optional) list of element of this number field

OUTPUT: integer if v is omitted, and Rational otherwise

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<i> = NumberField(x^2 + 1)
sage: K.discriminant()
-4
sage: K.<a> = NumberField(x^2 + 5)
sage: K.discriminant()
-20
sage: K.<a> = NumberField(x^2 - 5)
sage: K.discriminant()
5

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> K.discriminant()
-4
>>> K = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> K.discriminant()
-20
>>> K = NumberField(x**Integer(2) - Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> K.discriminant()
5

hilbert_class_field(names)[source]¶

Return the Hilbert class field of this quadratic field as a relative extension of this field.

Note

For the polynomial that defines this field as a relative extension, see the method hilbert_class_field_defining_polynomial(), which is vastly faster than this method, since it doesn’t construct a relative extension.

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^2 + 23)
sage: L = K.hilbert_class_field('b'); L
Number Field in b with defining polynomial x^3 - x^2 + 1 over its base field
sage: L.absolute_field('c')
Number Field in c with defining polynomial
 x^6 - 2*x^5 + 70*x^4 - 90*x^3 + 1631*x^2 - 1196*x + 12743
sage: K.hilbert_class_field_defining_polynomial()
x^3 - x^2 + 1

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = K._first_ngens(1)
>>> L = K.hilbert_class_field('b'); L
Number Field in b with defining polynomial x^3 - x^2 + 1 over its base field
>>> L.absolute_field('c')
Number Field in c with defining polynomial
 x^6 - 2*x^5 + 70*x^4 - 90*x^3 + 1631*x^2 - 1196*x + 12743
>>> K.hilbert_class_field_defining_polynomial()
x^3 - x^2 + 1

hilbert_class_field_defining_polynomial(name='x')[source]¶

Return a polynomial over \(\QQ\) whose roots generate the Hilbert class field of this quadratic field as an extension of this quadratic field.

Note

Computed using PARI via Schertz’s method. This implementation is quite fast.

EXAMPLES:

Sage

sage: K.<b> = QuadraticField(-23)
sage: K.hilbert_class_field_defining_polynomial()
x^3 - x^2 + 1

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(23), names=('b',)); (b,) = K._first_ngens(1)
>>> K.hilbert_class_field_defining_polynomial()
x^3 - x^2 + 1

Note that this polynomial is not the actual Hilbert class polynomial: see hilbert_class_polynomial:

Sage

sage: K.hilbert_class_polynomial()                                          # needs sage.schemes
x^3 + 3491750*x^2 - 5151296875*x + 12771880859375

Python

>>> from sage.all import *
>>> K.hilbert_class_polynomial()                                          # needs sage.schemes
x^3 + 3491750*x^2 - 5151296875*x + 12771880859375

Sage

sage: K.<a> = QuadraticField(-431)
sage: K.class_number()
21
sage: K.hilbert_class_field_defining_polynomial(name='z')
z^21 + 6*z^20 + 9*z^19 - 4*z^18 + 33*z^17 + 140*z^16 + 220*z^15 + 243*z^14
 + 297*z^13 + 461*z^12 + 658*z^11 + 743*z^10 + 722*z^9 + 681*z^8 + 619*z^7
 + 522*z^6 + 405*z^5 + 261*z^4 + 119*z^3 + 35*z^2 + 7*z + 1

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(431), names=('a',)); (a,) = K._first_ngens(1)
>>> K.class_number()
21
>>> K.hilbert_class_field_defining_polynomial(name='z')
z^21 + 6*z^20 + 9*z^19 - 4*z^18 + 33*z^17 + 140*z^16 + 220*z^15 + 243*z^14
 + 297*z^13 + 461*z^12 + 658*z^11 + 743*z^10 + 722*z^9 + 681*z^8 + 619*z^7
 + 522*z^6 + 405*z^5 + 261*z^4 + 119*z^3 + 35*z^2 + 7*z + 1

hilbert_class_polynomial(name='x')[source]¶

Compute the Hilbert class polynomial of this quadratic field.

Right now, this is only implemented for imaginary quadratic fields.

EXAMPLES:

Sage

sage: K.<a> = QuadraticField(-3)
sage: K.hilbert_class_polynomial()                                          # needs sage.schemes
x

sage: K.<a> = QuadraticField(-31)
sage: K.hilbert_class_polynomial(name='z')                                  # needs sage.schemes
z^3 + 39491307*z^2 - 58682638134*z + 1566028350940383

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> K.hilbert_class_polynomial()                                          # needs sage.schemes
x

>>> K = QuadraticField(-Integer(31), names=('a',)); (a,) = K._first_ngens(1)
>>> K.hilbert_class_polynomial(name='z')                                  # needs sage.schemes
z^3 + 39491307*z^2 - 58682638134*z + 1566028350940383

is_galois()[source]¶

Return True since all quadratic fields are automatically Galois.

EXAMPLES:

Sage

sage: QuadraticField(1234,'d').is_galois()
True

Python

>>> from sage.all import *
>>> QuadraticField(Integer(1234),'d').is_galois()
True

number_of_roots_of_unity()[source]¶

Return the number of roots of unity in this quadratic field.

This is always 2 except when \(d\) is \(-3\) or \(-4\).

EXAMPLES:

Sage

sage: QF = QuadraticField
sage: [QF(d).number_of_roots_of_unity() for d in range(-7, -2)]
[2, 2, 2, 4, 6]

Python

>>> from sage.all import *
>>> QF = QuadraticField
>>> [QF(d).number_of_roots_of_unity() for d in range(-Integer(7), -Integer(2))]
[2, 2, 2, 4, 6]

order_of_conductor(f)[source]¶

Return the unique order with the given conductor in this quadratic field.

EXAMPLES:

Sage

sage: K.<t> = QuadraticField(-123)
sage: K.order_of_conductor(1) is K.maximal_order()
True
sage: K.order_of_conductor(2).gens()
(1, t)
sage: K.order_of_conductor(44).gens()
(1, 22*t)
sage: K.order_of_conductor(9001).conductor()
9001

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(123), names=('t',)); (t,) = K._first_ngens(1)
>>> K.order_of_conductor(Integer(1)) is K.maximal_order()
True
>>> K.order_of_conductor(Integer(2)).gens()
(1, t)
>>> K.order_of_conductor(Integer(44)).gens()
(1, 22*t)
>>> K.order_of_conductor(Integer(9001)).conductor()
9001

sage.rings.number_field.number_field.NumberField_quadratic_v1(poly, name, canonical_embedding=None)[source]¶

Used for unpickling old pickles.

EXAMPLES:

Sage

sage: from sage.rings.number_field.number_field import NumberField_quadratic_v1
sage: R.<x> = QQ[]
sage: NumberField_quadratic_v1(x^2 - 2, 'd')
Number Field in d with defining polynomial x^2 - 2

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field import NumberField_quadratic_v1
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> NumberField_quadratic_v1(x**Integer(2) - Integer(2), 'd')
Number Field in d with defining polynomial x^2 - 2

sage.rings.number_field.number_field.QuadraticField(D, name='a', check=True, embedding=True, latex_name='sqrt', **args)[source]¶

Return a quadratic field obtained by adjoining a square root of \(D\) to the rational numbers, where \(D\) is not a perfect square.

INPUT:

D – a rational number
name – variable name (default: 'a')
check – boolean (default: True)
embedding – boolean or square root of \(D\) in an ambient field (default: True)
latex_name – latex variable name (default: \(\sqrt{D}\))

OUTPUT: a number field defined by a quadratic polynomial. Unless otherwise specified, it has an embedding into \(\RR\) or \(\CC\) by sending the generator to the positive or upper-half-plane root.

EXAMPLES:

Sage

sage: QuadraticField(3, 'a')
Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878?
sage: K.<theta> = QuadraticField(3); K
Number Field in theta with defining polynomial x^2 - 3 with theta = 1.732050807568878?
sage: RR(theta)
1.73205080756888
sage: QuadraticField(9, 'a')
Traceback (most recent call last):
...
ValueError: D must not be a perfect square.
sage: QuadraticField(9, 'a', check=False)
Number Field in a with defining polynomial x^2 - 9 with a = 3

Python

>>> from sage.all import *
>>> QuadraticField(Integer(3), 'a')
Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878?
>>> K = QuadraticField(Integer(3), names=('theta',)); (theta,) = K._first_ngens(1); K
Number Field in theta with defining polynomial x^2 - 3 with theta = 1.732050807568878?
>>> RR(theta)
1.73205080756888
>>> QuadraticField(Integer(9), 'a')
Traceback (most recent call last):
...
ValueError: D must not be a perfect square.
>>> QuadraticField(Integer(9), 'a', check=False)
Number Field in a with defining polynomial x^2 - 9 with a = 3

Quadratic number fields derive from general number fields.

Sage

sage: from sage.rings.number_field.number_field_base import NumberField
sage: type(K)
<class 'sage.rings.number_field.number_field.NumberField_quadratic_with_category'>
sage: isinstance(K, NumberField)
True

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field_base import NumberField
>>> type(K)
<class 'sage.rings.number_field.number_field.NumberField_quadratic_with_category'>
>>> isinstance(K, NumberField)
True

Quadratic number fields are cached:

Sage

sage: QuadraticField(-11, 'a') is QuadraticField(-11, 'a')
True

Python

>>> from sage.all import *
>>> QuadraticField(-Integer(11), 'a') is QuadraticField(-Integer(11), 'a')
True

By default, quadratic fields come with a nice latex representation:

Sage

sage: K.<a> = QuadraticField(-7)
sage: latex(K)
\Bold{Q}(\sqrt{-7})
sage: latex(a)
\sqrt{-7}
sage: latex(1/(1+a))
-\frac{1}{8} \sqrt{-7} + \frac{1}{8}
sage: list(K.latex_variable_names())
['\\sqrt{-7}']

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> latex(K)
\Bold{Q}(\sqrt{-7})
>>> latex(a)
\sqrt{-7}
>>> latex(Integer(1)/(Integer(1)+a))
-\frac{1}{8} \sqrt{-7} + \frac{1}{8}
>>> list(K.latex_variable_names())
['\\sqrt{-7}']

We can provide our own name as well:

Sage

sage: K.<a> = QuadraticField(next_prime(10^10), latex_name=r'\sqrt{D}')
sage: 1 + a
a + 1
sage: latex(1 + a)
\sqrt{D} + 1
sage: latex(QuadraticField(-1, 'a', latex_name=None).gen())
a

Python

>>> from sage.all import *
>>> K = QuadraticField(next_prime(Integer(10)**Integer(10)), latex_name=r'\sqrt{D}', names=('a',)); (a,) = K._first_ngens(1)
>>> Integer(1) + a
a + 1
>>> latex(Integer(1) + a)
\sqrt{D} + 1
>>> latex(QuadraticField(-Integer(1), 'a', latex_name=None).gen())
a

The name of the generator does not interfere with Sage preparser, see Issue #1135:

Sage

sage: K1 = QuadraticField(5, 'x')
sage: K2.<x> = QuadraticField(5)
sage: K3.<x> = QuadraticField(5, 'x')
sage: K1 is K2
True
sage: K1 is K3
True
sage: K1
Number Field in x with defining polynomial x^2 - 5 with x = 2.236067977499790?

Python

>>> from sage.all import *
>>> K1 = QuadraticField(Integer(5), 'x')
>>> K2 = QuadraticField(Integer(5), names=('x',)); (x,) = K2._first_ngens(1)
>>> K3 = QuadraticField(Integer(5), 'x', names=('x',)); (x,) = K3._first_ngens(1)
>>> K1 is K2
True
>>> K1 is K3
True
>>> K1
Number Field in x with defining polynomial x^2 - 5 with x = 2.236067977499790?

Note that, in presence of two different names for the generator, the name given by the preparser takes precedence:

Sage

sage: K4.<y> = QuadraticField(5, 'x'); K4
Number Field in y with defining polynomial x^2 - 5 with y = 2.236067977499790?
sage: K1 == K4
False

Python

>>> from sage.all import *
>>> K4 = QuadraticField(Integer(5), 'x', names=('y',)); (y,) = K4._first_ngens(1); K4
Number Field in y with defining polynomial x^2 - 5 with y = 2.236067977499790?
>>> K1 == K4
False

sage.rings.number_field.number_field.is_AbsoluteNumberField(x)[source]¶

Return True if x is an absolute number field.

EXAMPLES:

Sage

sage: from sage.rings.number_field.number_field import is_AbsoluteNumberField
sage: x = polygen(ZZ, 'x')
sage: is_AbsoluteNumberField(NumberField(x^2 + 1, 'a'))
doctest:warning...
DeprecationWarning: The function is_AbsoluteNumberField is deprecated; use 'isinstance(..., NumberField_absolute)' instead.
See https://github.com/sagemath/sage/issues/38124 for details.
True
sage: is_AbsoluteNumberField(NumberField([x^3 + 17, x^2 + 1], 'a'))
False

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field import is_AbsoluteNumberField
>>> x = polygen(ZZ, 'x')
>>> is_AbsoluteNumberField(NumberField(x**Integer(2) + Integer(1), 'a'))
doctest:warning...
DeprecationWarning: The function is_AbsoluteNumberField is deprecated; use 'isinstance(..., NumberField_absolute)' instead.
See https://github.com/sagemath/sage/issues/38124 for details.
True
>>> is_AbsoluteNumberField(NumberField([x**Integer(3) + Integer(17), x**Integer(2) + Integer(1)], 'a'))
False

The rationals are a number field, but they’re not of the absolute number field class.

Sage

sage: is_AbsoluteNumberField(QQ)
False

Python

>>> from sage.all import *
>>> is_AbsoluteNumberField(QQ)
False

sage.rings.number_field.number_field.is_NumberFieldHomsetCodomain(codomain)[source]¶

Return whether codomain is a valid codomain for a number field homset. This is used by NumberField._Hom_ to determine whether the created homsets should be a sage.rings.number_field.homset.NumberFieldHomset.

EXAMPLES:

This currently accepts any parent (CC, RR, …) in Fields:

Sage

sage: from sage.rings.number_field.number_field import is_NumberFieldHomsetCodomain
sage: is_NumberFieldHomsetCodomain(QQ)
True
sage: x = polygen(ZZ, 'x')
sage: is_NumberFieldHomsetCodomain(NumberField(x^2 + 1, 'x'))
True
sage: is_NumberFieldHomsetCodomain(ZZ)
False
sage: is_NumberFieldHomsetCodomain(3)
False
sage: is_NumberFieldHomsetCodomain(MatrixSpace(QQ, 2))
False
sage: is_NumberFieldHomsetCodomain(InfinityRing)
False

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field import is_NumberFieldHomsetCodomain
>>> is_NumberFieldHomsetCodomain(QQ)
True
>>> x = polygen(ZZ, 'x')
>>> is_NumberFieldHomsetCodomain(NumberField(x**Integer(2) + Integer(1), 'x'))
True
>>> is_NumberFieldHomsetCodomain(ZZ)
False
>>> is_NumberFieldHomsetCodomain(Integer(3))
False
>>> is_NumberFieldHomsetCodomain(MatrixSpace(QQ, Integer(2)))
False
>>> is_NumberFieldHomsetCodomain(InfinityRing)
False

Question: should, for example, QQ-algebras be accepted as well?

Caveat: Gap objects are not (yet) in Fields, and therefore not accepted as number field homset codomains:

Sage

sage: is_NumberFieldHomsetCodomain(gap.Rationals)                               # needs sage.libs.gap
False

Python

>>> from sage.all import *
>>> is_NumberFieldHomsetCodomain(gap.Rationals)                               # needs sage.libs.gap
False

sage.rings.number_field.number_field.is_fundamental_discriminant(D)[source]¶

Return True if the integer \(D\) is a fundamental discriminant, i.e., if \(D \cong 0,1\pmod{4}\), and \(D\neq 0, 1\) and either (1) \(D\) is square free or (2) we have \(D\cong 0\pmod{4}\) with \(D/4 \cong 2,3\pmod{4}\) and \(D/4\) square free. These are exactly the discriminants of quadratic fields.

EXAMPLES:

Sage

sage: [D for D in range(-15,15) if is_fundamental_discriminant(D)]
...
DeprecationWarning: is_fundamental_discriminant(D) is deprecated;
please use D.is_fundamental_discriminant()
...
[-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]
sage: [D for D in range(-15,15)
....:  if not is_square(D) and QuadraticField(D,'a').disc() == D]
[-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]

Python

>>> from sage.all import *
>>> [D for D in range(-Integer(15),Integer(15)) if is_fundamental_discriminant(D)]
...
DeprecationWarning: is_fundamental_discriminant(D) is deprecated;
please use D.is_fundamental_discriminant()
...
[-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]
>>> [D for D in range(-Integer(15),Integer(15))
...  if not is_square(D) and QuadraticField(D,'a').disc() == D]
[-15, -11, -8, -7, -4, -3, 5, 8, 12, 13]

sage.rings.number_field.number_field.is_real_place(v)[source]¶

Return True if \(v\) is real, False if \(v\) is complex.

INPUT:

v – an infinite place of self

OUTPUT:

A boolean indicating whether a place is real (True) or complex (False).

EXAMPLES:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<xi> = NumberField(x^3 - 3)
sage: phi_real = K.places()[0]
sage: phi_complex = K.places()[1]
sage: v_fin = tuple(K.primes_above(3))[0]

sage: is_real_place(phi_real)
True

sage: is_real_place(phi_complex)
False

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(3), names=('xi',)); (xi,) = K._first_ngens(1)
>>> phi_real = K.places()[Integer(0)]
>>> phi_complex = K.places()[Integer(1)]
>>> v_fin = tuple(K.primes_above(Integer(3)))[Integer(0)]

>>> is_real_place(phi_real)
True

>>> is_real_place(phi_complex)
False

It is an error to put in a finite place

Sage

sage: is_real_place(v_fin)
Traceback (most recent call last):
...
AttributeError: 'NumberFieldFractionalIdeal' object has no attribute 'im_gens'...

Python

>>> from sage.all import *
>>> is_real_place(v_fin)
Traceback (most recent call last):
...
AttributeError: 'NumberFieldFractionalIdeal' object has no attribute 'im_gens'...

sage.rings.number_field.number_field.proof_flag(t)[source]¶

Used for easily determining the correct proof flag to use.

Return t if t is not None, otherwise return the system-wide proof-flag for number fields (default: True).

EXAMPLES:

Sage

sage: from sage.rings.number_field.number_field import proof_flag
sage: proof_flag(True)
True
sage: proof_flag(False)
False
sage: proof_flag(None)
True
sage: proof_flag("banana")
'banana'

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field import proof_flag
>>> proof_flag(True)
True
>>> proof_flag(False)
False
>>> proof_flag(None)
True
>>> proof_flag("banana")
'banana'

sage.rings.number_field.number_field.put_natural_embedding_first(v)[source]¶

Helper function for embeddings() functions for number fields.

INPUT:

v – list of embeddings of a number field

OUTPUT: none; the list is altered in-place, so that, if possible, the first embedding has been switched with one of the others, so that if there is an embedding which preserves the generator names then it appears first.

EXAMPLES:

Sage

sage: K.<a> = CyclotomicField(7)
sage: embs = K.embeddings(K)
sage: [e(a) for e in embs]  # random - there is no natural sort order
[a, a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1]
sage: id = [e for e in embs if e(a) == a][0]; id
Ring endomorphism of Cyclotomic Field of order 7 and degree 6
  Defn: a |--> a
sage: permuted_embs = list(embs); permuted_embs.remove(id); permuted_embs.append(id)
sage: [e(a) for e in permuted_embs]  # random - but natural map is not first
[a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a]
sage: permuted_embs[0] != a
True
sage: from sage.rings.number_field.number_field import put_natural_embedding_first
sage: put_natural_embedding_first(permuted_embs)
sage: [e(a) for e in permuted_embs]  # random - but natural map is first
[a, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a^2]
sage: permuted_embs[0] == id
True

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(7), names=('a',)); (a,) = K._first_ngens(1)
>>> embs = K.embeddings(K)
>>> [e(a) for e in embs]  # random - there is no natural sort order
[a, a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1]
>>> id = [e for e in embs if e(a) == a][Integer(0)]; id
Ring endomorphism of Cyclotomic Field of order 7 and degree 6
  Defn: a |--> a
>>> permuted_embs = list(embs); permuted_embs.remove(id); permuted_embs.append(id)
>>> [e(a) for e in permuted_embs]  # random - but natural map is not first
[a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a]
>>> permuted_embs[Integer(0)] != a
True
>>> from sage.rings.number_field.number_field import put_natural_embedding_first
>>> put_natural_embedding_first(permuted_embs)
>>> [e(a) for e in permuted_embs]  # random - but natural map is first
[a, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a^2]
>>> permuted_embs[Integer(0)] == id
True

sage.rings.number_field.number_field.refine_embedding(e, prec=None)[source]¶

Given an embedding from a number field to either \(\RR\) or \(\CC\), return an equivalent embedding with higher precision.

INPUT:

e – an embedding of a number field into either \(\RR\) or \(\CC\) (with some precision)
prec – (default: None) the desired precision; if None, current precision is doubled; if Infinity, the equivalent embedding into either QQbar or AA is returned.

EXAMPLES:

Sage

sage: from sage.rings.number_field.number_field import refine_embedding
sage: K = CyclotomicField(3)
sage: e10 = K.complex_embedding(10)
sage: e10.codomain().precision()
10
sage: e25 = refine_embedding(e10, prec=25)
sage: e25.codomain().precision()
25

Python

>>> from sage.all import *
>>> from sage.rings.number_field.number_field import refine_embedding
>>> K = CyclotomicField(Integer(3))
>>> e10 = K.complex_embedding(Integer(10))
>>> e10.codomain().precision()
10
>>> e25 = refine_embedding(e10, prec=Integer(25))
>>> e25.codomain().precision()
25

An example where we extend a real embedding into AA:

Sage

sage: x = polygen(QQ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: K.signature()
(1, 1)
sage: e = K.embeddings(RR)[0]; e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Real Field with 53 bits of precision
Defn: a |--> 1.25992104989487
sage: e = refine_embedding(e, Infinity); e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Algebraic Real Field
Defn: a |--> 1.259921049894873?

Python

>>> from sage.all import *
>>> x = polygen(QQ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> K.signature()
(1, 1)
>>> e = K.embeddings(RR)[Integer(0)]; e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Real Field with 53 bits of precision
Defn: a |--> 1.25992104989487
>>> e = refine_embedding(e, Infinity); e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Algebraic Real Field
Defn: a |--> 1.259921049894873?

Now we can obtain arbitrary precision values with no trouble:

Sage

sage: RealField(150)(e(a))
1.2599210498948731647672106072782283505702515
sage: _^3
2.0000000000000000000000000000000000000000000
sage: RealField(200)(e(a^2 - 3*a + 7))
4.8076379022835799804500738174376232086807389337953290695624

Python

>>> from sage.all import *
>>> RealField(Integer(150))(e(a))
1.2599210498948731647672106072782283505702515
>>> _**Integer(3)
2.0000000000000000000000000000000000000000000
>>> RealField(Integer(200))(e(a**Integer(2) - Integer(3)*a + Integer(7)))
4.8076379022835799804500738174376232086807389337953290695624

Complex embeddings can be extended into QQbar:

Sage

sage: e = K.embeddings(CC)[0]; e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Complex Field with 53 bits of precision
Defn: a |--> -0.62996052494743... - 1.09112363597172*I
sage: e = refine_embedding(e,Infinity); e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Algebraic Field
Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I
sage: ComplexField(200)(e(a))
-0.62996052494743658238360530363911417528512573235075399004099
 - 1.0911236359717214035600726141898088813258733387403009407036*I
sage: e(a)^3
2

Python

>>> from sage.all import *
>>> e = K.embeddings(CC)[Integer(0)]; e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Complex Field with 53 bits of precision
Defn: a |--> -0.62996052494743... - 1.09112363597172*I
>>> e = refine_embedding(e,Infinity); e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Algebraic Field
Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I
>>> ComplexField(Integer(200))(e(a))
-0.62996052494743658238360530363911417528512573235075399004099
 - 1.0911236359717214035600726141898088813258733387403009407036*I
>>> e(a)**Integer(3)
2

Embeddings into lazy fields work:

Sage

sage: L = CyclotomicField(7)
sage: x = L.specified_complex_embedding(); x
Generic morphism:
  From: Cyclotomic Field of order 7 and degree 6
  To:   Complex Lazy Field
  Defn: zeta7 -> 0.623489801858734? + 0.781831482468030?*I
sage: refine_embedding(x, 300)
Ring morphism:
  From: Cyclotomic Field of order 7 and degree 6
  To:   Complex Field with 300 bits of precision
  Defn: zeta7 |--> 0.623489801858733530525004884004239810632274730896402105365549439096853652456487284575942507
                    + 0.781831482468029808708444526674057750232334518708687528980634958045091731633936441700868007*I
sage: refine_embedding(x, infinity)
Ring morphism:
  From: Cyclotomic Field of order 7 and degree 6
  To:   Algebraic Field
  Defn: zeta7 |--> 0.6234898018587335? + 0.7818314824680299?*I

Python

>>> from sage.all import *
>>> L = CyclotomicField(Integer(7))
>>> x = L.specified_complex_embedding(); x
Generic morphism:
  From: Cyclotomic Field of order 7 and degree 6
  To:   Complex Lazy Field
  Defn: zeta7 -> 0.623489801858734? + 0.781831482468030?*I
>>> refine_embedding(x, Integer(300))
Ring morphism:
  From: Cyclotomic Field of order 7 and degree 6
  To:   Complex Field with 300 bits of precision
  Defn: zeta7 |--> 0.623489801858733530525004884004239810632274730896402105365549439096853652456487284575942507
                    + 0.781831482468029808708444526674057750232334518708687528980634958045091731633936441700868007*I
>>> refine_embedding(x, infinity)
Ring morphism:
  From: Cyclotomic Field of order 7 and degree 6
  To:   Algebraic Field
  Defn: zeta7 |--> 0.6234898018587335? + 0.7818314824680299?*I

When the old embedding is into the real lazy field, then only real embeddings should be considered. See Issue #17495:

Sage

sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^3 + x - 1, embedding=0.68)
sage: from sage.rings.number_field.number_field import refine_embedding
sage: refine_embedding(K.specified_complex_embedding(), 100)
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + x - 1 with a = 0.6823278038280193?
  To:   Real Field with 100 bits of precision
  Defn: a |--> 0.68232780382801932736948373971
sage: refine_embedding(K.specified_complex_embedding(), Infinity)
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + x - 1 with a = 0.6823278038280193?
  To:   Algebraic Real Field
  Defn: a |--> 0.6823278038280193?

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> K = NumberField(x**Integer(3) + x - Integer(1), embedding=RealNumber('0.68'), names=('a',)); (a,) = K._first_ngens(1)
>>> from sage.rings.number_field.number_field import refine_embedding
>>> refine_embedding(K.specified_complex_embedding(), Integer(100))
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + x - 1 with a = 0.6823278038280193?
  To:   Real Field with 100 bits of precision
  Defn: a |--> 0.68232780382801932736948373971
>>> refine_embedding(K.specified_complex_embedding(), Infinity)
Ring morphism:
  From: Number Field in a with defining polynomial x^3 + x - 1 with a = 0.6823278038280193?
  To:   Algebraic Real Field
  Defn: a |--> 0.6823278038280193?