Relative Number Fields¶
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
Nick Alexander (2009-01): modernize coercion implementation
Robert Harron (2012-08): added is_CM_extension
Julian Rueth (2014-04-03): absolute number fields are unique parents
This example follows one in the Magma reference manual:
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
We do some arithmetic in a tower of relative number fields:
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
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).
- sage.rings.number_field.number_field_rel.NumberField_extension_v1(base_field, poly, name, latex_name, canonical_embedding=None)¶
Used for unpickling old pickles.
EXAMPLES:
sage: from sage.rings.number_field.number_field_rel import NumberField_relative_v1 sage: R.<x> = CyclotomicField(3)[] sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a') Number Field in a with defining polynomial x^2 + 7 over its base field
- class sage.rings.number_field.number_field_rel.NumberField_relative(base, polynomial, name, latex_name=None, names=None, check=True, embedding=None, structure=None)¶
Bases:
sage.rings.number_field.number_field.NumberField_generic
INPUT:
base
– the base fieldpolynomial
– a polynomial which must be defined in the ring \(K[x]\), where \(K\) is the base field.name
– a string, the variable namelatex_name
– a string orNone
(default:None
), variable name for latex printingcheck
– a boolean (default:True
), whether to check irreducibility ofpolynomial
embedding
– currently not supported, must beNone
structure
– an instance ofstructure.NumberFieldStructure
orNone
(default:None
), provides additional information about this number field, e.g., the absolute number field from which it was created
EXAMPLES:
sage: K.<a> = NumberField(x^3 - 2) sage: t = polygen(K) sage: L.<b> = K.extension(t^2+t+a); L Number Field in b with defining polynomial x^2 + x + a over its base field
- absolute_base_field()¶
Return the base field of this relative extension, but viewed as an absolute field over \(\QQ\).
EXAMPLES:
sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 3, x^3 + 2]) sage: K Number Field in a with defining polynomial x^2 + 2 over its base field sage: K.base_field() Number Field in b with defining polynomial x^3 + 3 over its base field sage: K.absolute_base_field()[0] Number Field in a0 with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1 sage: K.base_field().absolute_field('z') Number Field in z with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1
- absolute_degree()¶
The degree of this relative number field over the rational field.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2]) sage: K.absolute_degree() 6
- absolute_different()¶
Return the absolute different of this relative number field \(L\), as an ideal of \(L\). To get the relative different of \(L/K\), use
L.relative_different()
.EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1) sage: t = K['t'].gen() sage: L.<b> = K.extension(t^4 - i) sage: L.absolute_different() Fractional ideal (8)
- absolute_discriminant(v=None)¶
Return the absolute discriminant of this relative number field or if
v
is specified, the determinant of the trace pairing on the elements of the listv
.INPUT:
v
(optional) – list of element of this relative number field.
OUTPUT: Integer if
v
is omitted, and Rational otherwise.EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1) sage: t = K['t'].gen() sage: L.<b> = K.extension(t^4 - i) sage: L.absolute_discriminant() 16777216 sage: L.absolute_discriminant([(b + i)^j for j in range(8)]) 61911970349056
- absolute_field(names)¶
Return
self
as an absolute number field.INPUT:
names
– string; name of generator of the absolute field
OUTPUT:
An absolute number field \(K\) that is isomorphic to this field.
Also,
K.structure()
returnsfrom_K
andto_K
, wherefrom_K
is an isomorphism from \(K\) toself
andto_K
is an isomorphism fromself
to \(K\).EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: L.<xyz> = K.absolute_field(); L Number Field in xyz with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49 sage: L.<c> = K.absolute_field(); L Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49 sage: from_L, to_L = L.structure() sage: from_L Isomorphism map: From: Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49 To: Number Field in a with defining polynomial x^4 + 3 over its base field sage: from_L(c) a - b sage: to_L Isomorphism map: From: Number Field in a with defining polynomial x^4 + 3 over its base field To: Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49 sage: to_L(a) -5/182*c^7 - 87/364*c^5 - 185/182*c^3 + 323/364*c sage: to_L(b) -5/182*c^7 - 87/364*c^5 - 185/182*c^3 - 41/364*c sage: to_L(a)^4 -3 sage: to_L(b)^2 -2
- absolute_generator()¶
Return the chosen generator over \(\QQ\) for this relative number field.
EXAMPLES:
sage: y = polygen(QQ,'y') sage: k.<a> = NumberField([y^2 + 2, y^4 + 3]) sage: g = k.absolute_generator(); g a0 - a1 sage: g.minpoly() x^2 + 2*a1*x + a1^2 + 2 sage: g.absolute_minpoly() x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
- absolute_polynomial()¶
Return the polynomial over \(\QQ\) that defines this field as an extension of the rational numbers.
Note
The absolute polynomial of a relative number field is chosen to be equal to the defining polynomial of the underlying PARI absolute number field (it cannot be specified by the user). In particular, it is always a monic polynomial with integral coefficients. On the other hand, the defining polynomial of an absolute number field and the relative polynomial of a relative number field are in general different from their PARI counterparts.
EXAMPLES:
sage: k.<a, b> = NumberField([x^2 + 1, x^3 + x + 1]); k Number Field in a with defining polynomial x^2 + 1 over its base field sage: k.absolute_polynomial() x^6 + 5*x^4 - 2*x^3 + 4*x^2 + 4*x + 1
An example comparing the various defining polynomials to their PARI counterparts:
sage: k.<a, c> = NumberField([x^2 + 1/3, x^2 + 1/4]) sage: k.absolute_polynomial() x^4 - x^2 + 1 sage: k.pari_polynomial() x^4 - x^2 + 1 sage: k.base_field().absolute_polynomial() x^2 + 1/4 sage: k.pari_absolute_base_polynomial() y^2 + 1 sage: k.relative_polynomial() x^2 + 1/3 sage: k.pari_relative_polynomial() x^2 + Mod(y, y^2 + 1)*x - 1
- absolute_polynomial_ntl()¶
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: NumberField(x^2 + (2/3)*x - 9/17,'a').absolute_polynomial_ntl() ([-27 34 51], 51)
- absolute_vector_space(base=None, *args, **kwds)¶
Return vector space over \(\QQ\) of
self
and isomorphisms from the vector space toself
and in the other direction.EXAMPLES:
sage: K.<a,b> = NumberField([x^3 + 3, x^3 + 2]); K Number Field in a with defining polynomial x^3 + 3 over its base field sage: V,from_V,to_V = K.absolute_vector_space(); V Vector space of dimension 9 over Rational Field sage: from_V Isomorphism map: From: Vector space of dimension 9 over Rational Field To: Number Field in a with defining polynomial x^3 + 3 over its base field sage: to_V Isomorphism map: From: Number Field in a with defining polynomial x^3 + 3 over its base field To: Vector space of dimension 9 over Rational Field sage: c = (a+1)^5; c 7*a^2 - 10*a - 29 sage: to_V(c) (-29, -712/9, 19712/45, 0, -14/9, 364/45, 0, -4/9, 119/45) sage: from_V(to_V(c)) 7*a^2 - 10*a - 29 sage: from_V(3*to_V(b)) 3*b
- automorphisms()¶
Compute all Galois automorphisms of self over the base field. This is different than computing the embeddings of self into self; there, automorphisms that do not fix the base field are considered.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + 10000, x^2 + x + 50]); K Number Field in a with defining polynomial x^2 + 10000 over its base field sage: K.automorphisms() [ Relative number field endomorphism of Number Field in a with defining polynomial x^2 + 10000 over its base field Defn: a |--> a b |--> b, Relative number field endomorphism of Number Field in a with defining polynomial x^2 + 10000 over its base field Defn: a |--> -a b |--> b ] sage: rho, tau = K.automorphisms() sage: tau(a) -a sage: tau(b) == b True sage: L.<b, a> = NumberField([x^2 + x + 50, x^2 + 10000, ]); L Number Field in b with defining polynomial x^2 + x + 50 over its base field sage: L.automorphisms() [ Relative number field endomorphism of Number Field in b with defining polynomial x^2 + x + 50 over its base field Defn: b |--> b a |--> a, Relative number field endomorphism of Number Field in b with defining polynomial x^2 + x + 50 over its base field Defn: b |--> -b - 1 a |--> a ] sage: rho, tau = L.automorphisms() sage: tau(a) == a True sage: tau(b) -b - 1 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: K.automorphisms() [ Relative number field endomorphism of Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field Defn: c |--> c a |--> a b |--> b, Relative number field endomorphism of Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field Defn: c |--> -c a |--> a b |--> b ]
- base_field()¶
Return the base field of this relative number field.
EXAMPLES:
sage: k.<a> = NumberField([x^3 + x + 1]) sage: R.<z> = k[] sage: L.<b> = NumberField(z^3 + a) sage: L.base_field() Number Field in a with defining polynomial x^3 + x + 1 sage: L.base_field() is k True
This is very useful because the print representation of a relative field doesn’t describe the base field.:
sage: L Number Field in b with defining polynomial z^3 + a over its base field
- base_ring()¶
This is exactly the same as base_field.
EXAMPLES:
sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1]) sage: k.base_ring() Number Field in a1 with defining polynomial x^3 + x + 1 sage: k.base_field() Number Field in a1 with defining polynomial x^3 + x + 1
- change_names(names)¶
Return relative number field isomorphic to self but with the given generator names.
INPUT:
names
– number of names should be at most the number of generators of self, i.e., the number of steps in the tower of relative fields.
Also,
K.structure()
returnsfrom_K
andto_K
, where from_K is an isomorphism from \(K\) to self andto_K
is an isomorphism from self to \(K\).EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: L.<c,d> = K.change_names() sage: L Number Field in c with defining polynomial x^4 + 3 over its base field sage: L.base_field() Number Field in d with defining polynomial x^2 + 2
An example with a 3-level tower:
sage: K.<a,b,c> = NumberField([x^2 + 17, x^2 + x + 1, x^3 - 2]); K Number Field in a with defining polynomial x^2 + 17 over its base field sage: L.<m,n,r> = K.change_names() sage: L Number Field in m with defining polynomial x^2 + 17 over its base field sage: L.base_field() Number Field in n with defining polynomial x^2 + x + 1 over its base field sage: L.base_field().base_field() Number Field in r with defining polynomial x^3 - 2
And a more complicated example:
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: L.<m, n, r> = K.change_names(); L Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field sage: L.structure() (Isomorphism given by variable name change map: From: Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field To: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field, Isomorphism given by variable name change map: From: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field To: Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field)
- composite_fields(other, names=None, both_maps=False, preserve_embedding=True)¶
List of all possible composite number fields formed from self and other, together with (optionally) embeddings into the compositum; see the documentation for both_maps below.
Since relative fields do not have ambient embeddings, preserve_embedding has no effect. In every case all possible composite number fields are returned.
INPUT:
other
- a number fieldnames
- generator name for composite fieldsboth_maps
- (default: False) if True, return quadruples (F, self_into_F, other_into_F, k) such that self_into_F maps self into F, other_into_F maps other into F. For relative number fields k is always None.preserve_embedding
- (default: True) has no effect, but is kept for compatibility with the absolute version of this function. In every case the list of all possible compositums is returned.
OUTPUT:
list
- list of the composite fields, possibly with maps.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + 5, x^2 - 2]) sage: L.<c, d> = NumberField([x^2 + 5, x^2 - 3]) sage: K.composite_fields(L, 'e') [Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600] sage: K.composite_fields(L, 'e', both_maps=True) [[Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600, Relative number field morphism: From: Number Field in a with defining polynomial x^2 + 5 over its base field To: Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600 Defn: a |--> -9/66560*e^7 + 11/4160*e^5 - 241/4160*e^3 - 101/104*e b |--> -21/166400*e^7 + 73/20800*e^5 - 779/10400*e^3 + 7/260*e, Relative number field morphism: From: Number Field in c with defining polynomial x^2 + 5 over its base field To: Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600 Defn: c |--> -9/66560*e^7 + 11/4160*e^5 - 241/4160*e^3 - 101/104*e d |--> -3/25600*e^7 + 7/1600*e^5 - 147/1600*e^3 + 1/40*e, None]]
- defining_polynomial()¶
Return the defining polynomial of this relative number field.
This is exactly the same as
relative_polynomial()
.EXAMPLES:
sage: C.<z> = CyclotomicField(5) sage: PC.<X> = C[] sage: K.<a> = C.extension(X^2 + X + z); K Number Field in a with defining polynomial X^2 + X + z over its base field sage: K.defining_polynomial() X^2 + X + z
- degree()¶
The degree, unqualified, of a relative number field is deliberately not implemented, so that a user cannot mistake the absolute degree for the relative degree, or vice versa.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2]) sage: K.degree() Traceback (most recent call last): ... NotImplementedError: For a relative number field you must use relative_degree or absolute_degree as appropriate
- different()¶
The different, unqualified, of a relative number field is deliberately not implemented, so that a user cannot mistake the absolute different for the relative different, or vice versa.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1]) sage: K.different() Traceback (most recent call last): ... NotImplementedError: For a relative number field you must use relative_different or absolute_different as appropriate
- disc()¶
The discriminant, unqualified, of a relative number field is deliberately not implemented, so that a user cannot mistake the absolute discriminant for the relative discriminant, or vice versa.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1]) sage: K.disc() Traceback (most recent call last): ... NotImplementedError: For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate
- discriminant()¶
The discriminant, unqualified, of a relative number field is deliberately not implemented, so that a user cannot mistake the absolute discriminant for the relative discriminant, or vice versa.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1]) sage: K.discriminant() Traceback (most recent call last): ... NotImplementedError: For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate
- embeddings(K)¶
Compute all field embeddings of the relative number field self 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 self into \(K\) is put first in the list.
INPUT:
K
– a field
EXAMPLES:
sage: K.<a,b> = NumberField([x^3 - 2, x^2+1]) sage: f = K.embeddings(ComplexField(58)); f [ Relative number field morphism: From: Number Field in a with defining polynomial x^3 - 2 over its base field To: Complex Field with 58 bits of precision Defn: a |--> -0.62996052494743676 - 1.0911236359717214*I b |--> -1.9428902930940239e-16 + 1.0000000000000000*I, ... Relative number field morphism: From: Number Field in a with defining polynomial x^3 - 2 over its base field To: Complex Field with 58 bits of precision Defn: a |--> 1.2599210498948731 b |--> -0.99999999999999999*I ] sage: f[0](a)^3 2.0000000000000002 - 8.6389229103644993e-16*I sage: f[0](b)^2 -1.0000000000000001 - 3.8857805861880480e-16*I sage: f[0](a+b) -0.62996052494743693 - 0.091123635971721295*I
- free_module(base=None, basis=None, map=True)¶
Return a vector space over a specified subfield that is isomorphic to this number field, together with the isomorphisms in each direction.
INPUT:
base
– a subfieldbasis
– (optional) a list of elements giving a basis over the subfieldmap
– (defaultTrue
) whether to return isomorphisms to and from the vector space
EXAMPLES:
sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 2, x^3 + 3]); K Number Field in a with defining polynomial x^2 + 2 over its base field sage: V, from_V, to_V = K.free_module() sage: to_V(K.0) (0, 1) sage: W, from_W, to_W = K.free_module(base=QQ) sage: w = to_W(K.0); len(w) 18 sage: w[0] -127917622658689792301282/48787705559800061938765
- galois_closure(names=None)¶
Return the absolute number field \(K\) that is the Galois closure of this relative number field.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: K.galois_closure('c') Number Field in c with defining polynomial x^16 + 16*x^14 + 28*x^12 + 784*x^10 + 19846*x^8 - 595280*x^6 + 2744476*x^4 + 3212848*x^2 + 29953729
- gen(n=0)¶
Return the \(n\)’th generator of this relative number field.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: K.gens() (a, b) sage: K.gen(0) a
- gens()¶
Return the generators of this relative number field.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: K.gens() (a, b)
- is_CM_extension()¶
Return True is this is a CM extension, i.e. a totally imaginary quadratic extension of a totally real field.
EXAMPLES:
sage: F.<a> = NumberField(x^2 - 5) sage: K.<z> = F.extension(x^2 + 7) sage: K.is_CM_extension() True sage: K = CyclotomicField(7) sage: K_rel = K.relativize(K.gen() + K.gen()^(-1), 'z') sage: K_rel.is_CM_extension() True sage: F = CyclotomicField(3) sage: K.<z> = F.extension(x^3 - 2) sage: K.is_CM_extension() False
A CM field K such that K/F is not a CM extension
sage: F.<a> = NumberField(x^2 + 1) sage: K.<z> = F.extension(x^2 - 3) sage: K.is_CM_extension() False sage: K.is_CM() True
- is_absolute()¶
Returns False, since this is not an absolute field.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: K.is_absolute() False sage: K.is_relative() True
- is_free(proof=None)¶
Determine whether or not \(L/K\) is free (i.e. if \(\mathcal{O}_L\) is a free \(\mathcal{O}_K\)-module).
INPUT:
proof
– default: True
EXAMPLES:
sage: x = polygen(QQ) sage: K.<a> = NumberField(x^2+6) sage: x = polygen(K) sage: L.<b> = K.extension(x^2 + 3) ## extend by x^2+3 sage: L.is_free() False
- is_galois()¶
For a relative number field,
is_galois()
is deliberately not implemented, since it is not clear whether this would mean “Galois over \(\QQ\)” or “Galois over the given base field”. Use eitheris_galois_absolute()
oris_galois_relative()
respectively.EXAMPLES:
sage: k.<a> =NumberField([x^3 - 2, x^2 + x + 1]) sage: k.is_galois() Traceback (most recent call last): ... NotImplementedError: For a relative number field L you must use either L.is_galois_relative() or L.is_galois_absolute() as appropriate
- is_galois_absolute()¶
Return True if for this relative extension \(L/K\), \(L\) is a Galois extension of \(\QQ\).
EXAMPLES:
sage: K.<a> = NumberField(x^3 - 2) sage: y = polygen(K); L.<b> = K.extension(y^2 - a) sage: L.is_galois_absolute() False
- is_galois_relative()¶
Return True if for this relative extension \(L/K\), \(L\) is a Galois extension of \(K\).
EXAMPLES:
sage: K.<a> = NumberField(x^3 - 2) sage: y = polygen(K) sage: L.<b> = K.extension(y^2 - a) sage: L.is_galois_relative() True sage: M.<c> = K.extension(y^3 - a) sage: M.is_galois_relative() False
The next example previously gave a wrong result; see trac ticket #9390:
sage: F.<a, b> = NumberField([x^2 - 2, x^2 - 3]) sage: F.is_galois_relative() True
- is_isomorphic_relative(other, base_isom=None)¶
For this relative extension \(L/K\) and another relative extension \(M/K\), return True if there is a \(K\)-linear isomorphism from \(L\) to \(M\). More generally,
other
can be a relative extension \(M/K^\prime\) withbase_isom
an isomorphism from \(K\) to \(K^\prime\).EXAMPLES:
sage: K.<z9> = NumberField(x^6 + x^3 + 1) sage: R.<z> = PolynomialRing(K) sage: m1 = 3*z9^4 - 4*z9^3 - 4*z9^2 + 3*z9 - 8 sage: L1 = K.extension(z^2 - m1, 'b1') sage: G = K.galois_group(); gamma = G.gen() sage: m2 = (gamma^2)(m1) sage: L2 = K.extension(z^2 - m2, 'b2') sage: L1.is_isomorphic_relative(L2) False sage: L1.is_isomorphic(L2) True sage: L3 = K.extension(z^4 - m1, 'b3') sage: L1.is_isomorphic_relative(L3) False
If we have two extensions over different, but isomorphic, bases, we can compare them by letting
base_isom
be an isomorphism from self’s base field to other’s base field:sage: Kcyc.<zeta9> = CyclotomicField(9) sage: Rcyc.<zcyc> = PolynomialRing(Kcyc) sage: phi1 = K.hom([zeta9]) sage: m1cyc = phi1(m1) sage: L1cyc = Kcyc.extension(zcyc^2 - m1cyc, 'b1cyc') sage: L1.is_isomorphic_relative(L1cyc, base_isom=phi1) True sage: L2.is_isomorphic_relative(L1cyc, base_isom=phi1) False sage: phi2 = K.hom([phi1((gamma^(-2))(z9))]) sage: L1.is_isomorphic_relative(L1cyc, base_isom=phi2) False sage: L2.is_isomorphic_relative(L1cyc, base_isom=phi2) True
Omitting
base_isom
raises a ValueError when the base fields are not identical:sage: L1.is_isomorphic_relative(L1cyc) Traceback (most recent call last): ... ValueError: other does not have the same base field as self, so an isomorphism from self's base_field to other's base_field must be provided using the base_isom parameter.
The parameter
base_isom
can also be used to check if the relative extensions are Galois conjugate:sage: for g in G: ....: if L1.is_isomorphic_relative(L2, g.as_hom()): ....: print(g.as_hom()) Ring endomorphism of Number Field in z9 with defining polynomial x^6 + x^3 + 1 Defn: z9 |--> z9^4
- lift_to_base(element)¶
Lift an element of this extension into the base field if possible, or raise a ValueError if it is not possible.
EXAMPLES:
sage: x = polygen(ZZ) sage: K.<a> = NumberField(x^3 - 2) sage: R.<y> = K[] sage: L.<b> = K.extension(y^2 - a) sage: L.lift_to_base(b^4) a^2 sage: L.lift_to_base(b^6) 2 sage: L.lift_to_base(355/113) 355/113 sage: L.lift_to_base(b) Traceback (most recent call last): ... ValueError: The element b is not in the base field
- maximal_order(v=None)¶
Return the maximal order, i.e., the ring of integers of this number field.
INPUT:
v
- (default: None) None, a prime, or a list of primes.if v is None, return the maximal order.
if v is a prime, return an order that is p-maximal.
if v is a list, return an order that is maximal at each prime in the list v.
EXAMPLES:
sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3]) sage: OK = K.maximal_order(); 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
The following was previously “ridiculously slow”; see trac ticket #4738:
sage: K.<a,b> = NumberField([x^4 + 1, x^4 - 3]) sage: K.maximal_order() Maximal Relative Order in Number Field in a with defining polynomial x^4 + 1 over its base field
An example with nontrivial
v
:sage: L.<a,b> = NumberField([x^2 - 1000003, x^2 - 5*1000099^2]) sage: O3 = L.maximal_order([3]) sage: O3.absolute_discriminant() 400160824478095086350656915693814563600 sage: O3.is_maximal() False
- ngens()¶
Return the number of generators of this relative number field.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: K.gens() (a, b) sage: K.ngens() 2
- number_of_roots_of_unity()¶
Return number of roots of unity in this relative field.
EXAMPLES:
sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] ) sage: K.number_of_roots_of_unity() 24
- order(*gens, **kwds)¶
Return the order with given ring generators in the maximal order of this number field.
INPUT:
gens
– list of elements of self; if no generators are given, just returns the cardinality of this number field (oo) for consistency.check_is_integral
– bool (default: True), whether to check that each generator is integral.check_rank
– bool (default: True), whether to check that the ring generated by gens is of full rank.allow_subfield
– bool (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.
The check_is_integral and check_rank inputs must be given as explicit keyword arguments.
EXAMPLES:
sage: P.<a,b,c> = QQ[2^(1/2), 2^(1/3), 3^(1/2)] sage: R = P.order([a,b,c]); R Relative Order in Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
The base ring of an order in a relative extension is still \(\ZZ\).:
sage: R.base_ring() Integer Ring
One must give enough generators to generate a ring of finite index in the maximal order:
sage: P.order([a,b]) Traceback (most recent call last): ... ValueError: the rank of the span of gens is wrong
- pari_absolute_base_polynomial()¶
Return the PARI polynomial defining the absolute base field, in
y
.EXAMPLES:
sage: x = polygen(ZZ) sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 3]); K Number Field in a with defining polynomial x^2 + 2 over its base field sage: K.pari_absolute_base_polynomial() y^2 + 3 sage: type(K.pari_absolute_base_polynomial()) <type 'cypari2.gen.Gen'> sage: z = ZZ['z'].0 sage: K.<a, b, c> = NumberField([z^2 + 2, z^2 + 3, z^2 + 5]); K Number Field in a with defining polynomial z^2 + 2 over its base field sage: K.pari_absolute_base_polynomial() y^4 + 16*y^2 + 4 sage: K.base_field() Number Field in b with defining polynomial z^2 + 3 over its base field sage: len(QQ['y'](K.pari_absolute_base_polynomial()).roots(K.base_field())) 4 sage: type(K.pari_absolute_base_polynomial()) <type 'cypari2.gen.Gen'>
- pari_relative_polynomial()¶
Return the PARI relative polynomial associated to this number field.
This is always a polynomial in x and y, suitable for PARI’s rnfinit function. Notice that if this is a relative extension of a relative extension, the base field is the absolute base field.
EXAMPLES:
sage: k.<i> = NumberField(x^2 + 1) sage: m.<z> = k.extension(k['w']([i,0,1])) sage: m Number Field in z with defining polynomial w^2 + i over its base field sage: m.pari_relative_polynomial() Mod(1, y^2 + 1)*x^2 + Mod(y, y^2 + 1) sage: l.<t> = m.extension(m['t'].0^2 + z) sage: l.pari_relative_polynomial() Mod(1, y^4 + 1)*x^2 + Mod(y, y^4 + 1)
- pari_rnf()¶
Return the PARI relative number field object associated to this relative extension.
EXAMPLES:
sage: k.<a> = NumberField([x^4 + 3, x^2 + 2]) sage: k.pari_rnf() [x^4 + 3, [364, -10*x^7 - 87*x^5 - 370*x^3 - 41*x], [108, 3], ...]
- places(all_complex=False, prec=None)¶
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).
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: L.<b, c> = NumberFieldTower([x^2 - 5, x^3 + x + 3]) sage: L.places() [Relative number field morphism: From: Number Field in b with defining polynomial x^2 - 5 over its base field To: Real Field with 106 bits of precision Defn: b |--> -2.236067977499789696409173668937 c |--> -1.213411662762229634132131377426, Relative number field morphism: From: Number Field in b with defining polynomial x^2 - 5 over its base field To: Real Field with 106 bits of precision Defn: b |--> 2.236067977499789696411548005367 c |--> -1.213411662762229634130492421800, Relative number field morphism: From: Number Field in b with defining polynomial x^2 - 5 over its base field To: Complex Field with 53 bits of precision Defn: b |--> -2.23606797749979 ...e-1...*I c |--> 0.606705831381... - 1.45061224918844*I, Relative number field morphism: From: Number Field in b with defining polynomial x^2 - 5 over its base field To: Complex Field with 53 bits of precision Defn: b |--> 2.23606797749979 - 4.44089209850063e-16*I c |--> 0.606705831381115 - 1.45061224918844*I]
- polynomial()¶
For a relative number field,
polynomial()
is deliberately not implemented. Eitherrelative_polynomial()
orabsolute_polynomial()
must be used.EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1]) sage: K.polynomial() Traceback (most recent call last): ... NotImplementedError: For a relative number field L you must use either L.relative_polynomial() or L.absolute_polynomial() as appropriate
- relative_degree()¶
Returns the relative degree of this relative number field.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2]) sage: K.relative_degree() 2
- relative_different()¶
Return the relative different of this extension \(L/K\) as an ideal of \(L\). If you want the absolute different of \(L/\QQ\), use
L.absolute_different()
.EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1) sage: PK.<t> = K[] sage: L.<a> = K.extension(t^4 - i) sage: L.relative_different() Fractional ideal (4)
- relative_discriminant()¶
Return the relative discriminant of this extension \(L/K\) as an ideal of \(K\). If you want the (rational) discriminant of \(L/\QQ\), use e.g.
L.absolute_discriminant()
.EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1) sage: t = K['t'].gen() sage: L.<b> = K.extension(t^4 - i) sage: L.relative_discriminant() Fractional ideal (256) 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: K.relative_discriminant() == F.ideal(4*b) True
- relative_polynomial()¶
Return the defining polynomial of this relative number field over its base field.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1]) sage: K.relative_polynomial() x^2 + x + 1
Use absolute polynomial for a polynomial that defines the absolute extension.:
sage: K.absolute_polynomial() x^6 + 3*x^5 + 8*x^4 + 9*x^3 + 7*x^2 + 6*x + 3
- relative_vector_space(base=None, *args, **kwds)¶
Return vector space over the base field of self and isomorphisms from the vector space to self and in the other direction.
EXAMPLES:
sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 2, x^3 + 3]); K Number Field in a with defining polynomial x^2 + 2 over its base field sage: V, from_V, to_V = K.relative_vector_space() sage: from_V(V.0) 1 sage: to_V(K.0) (0, 1) sage: from_V(to_V(K.0)) a sage: to_V(from_V(V.0)) (1, 0) sage: to_V(from_V(V.1)) (0, 1)
The underlying vector space and maps is cached:
sage: W, from_V, to_V = K.relative_vector_space() sage: V is W True
- relativize(alpha, names)¶
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 selfnames
– name of generator for output field \(K\).
OUTPUT: \(K\) – a relative number field
Also,
K.structure()
returnsfrom_K
andto_K
, wherefrom_K
is an isomorphism from \(K\) to self andto_K
is an isomorphism from self to \(K\).EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K Number Field in a with defining polynomial x^4 + 3 over its base field sage: L.<z,w> = K.relativize(a^2) sage: z^2 z^2 sage: w^2 -3 sage: L Number Field in z with defining polynomial x^4 + (-2*w + 4)*x^2 + 4*w + 1 over its base field sage: L.base_field() Number Field in w with defining polynomial x^2 + 3
Now suppose we have \(K\) below \(L\) below \(M\):
sage: M = NumberField(x^8 + 2, 'a'); M Number Field in a with defining polynomial x^8 + 2 sage: L, L_into_M, _ = M.subfields(4)[0]; L Number Field in a0 with defining polynomial x^4 + 2 sage: K, K_into_L, _ = L.subfields(2)[0]; K Number Field in a0_0 with defining polynomial x^2 + 2 sage: K_into_M = L_into_M * K_into_L sage: L_over_K = L.relativize(K_into_L, 'c'); L_over_K Number Field in c with defining polynomial x^2 + a0_0 over its base field sage: L_over_K_to_L, L_to_L_over_K = L_over_K.structure() sage: M_over_L_over_K = M.relativize(L_into_M * L_over_K_to_L, 'd'); M_over_L_over_K Number Field in d with defining polynomial x^2 + c over its base field sage: M_over_L_over_K.base_field() is L_over_K True
Test relativizing a degree 6 field over its degree 2 and degree 3 subfields, using both an explicit element:
sage: K.<a> = NumberField(x^6 + 2); K Number Field in a with defining polynomial x^6 + 2 sage: K2, K2_into_K, _ = K.subfields(2)[0]; K2 Number Field in a0 with defining polynomial x^2 + 2 sage: K3, K3_into_K, _ = K.subfields(3)[0]; K3 Number Field in a0 with defining polynomial x^3 - 2
Here we explicitly relativize over an element of K2 (not the generator):
sage: L = K.relativize(K3_into_K, 'b'); L Number Field in b with defining polynomial x^2 + a0 over its base field sage: L_to_K, K_to_L = L.structure() sage: L_over_K2 = L.relativize(K_to_L(K2_into_K(K2.gen() + 1)), 'c'); L_over_K2 Number Field in c0 with defining polynomial x^3 - c1 + 1 over its base field sage: L_over_K2.base_field() Number Field in c1 with defining polynomial x^2 - 2*x + 3
Here we use a morphism to preserve the base field information:
sage: K2_into_L = K_to_L * K2_into_K sage: L_over_K2 = L.relativize(K2_into_L, 'c'); L_over_K2 Number Field in c with defining polynomial x^3 - a0 over its base field sage: L_over_K2.base_field() is K2 True
- roots_of_unity()¶
Return all the roots of unity in this relative field, primitive or not.
EXAMPLES:
sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] ) sage: rts = K.roots_of_unity() sage: len(rts) 24 sage: all(u in rts for u in [b*a, -b^2*a - b^2, b^3, -a, b*a + b]) True
- subfields(degree=0, name=None)¶
Return all subfields of this relative number field 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: 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: K.subfields(2) [ (Number Field in c0 with defining polynomial x^2 - 24*x + 96, Ring morphism: From: Number Field in c0 with defining polynomial x^2 - 24*x + 96 To: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field Defn: c0 |--> -4*b + 12, None), (Number Field in c1 with defining polynomial x^2 - 24*x + 120, Ring morphism: From: Number Field in c1 with defining polynomial x^2 - 24*x + 120 To: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field Defn: c1 |--> 2*b*a + 12, None), (Number Field in c2 with defining polynomial x^2 - 24*x + 72, Ring morphism: From: Number Field in c2 with defining polynomial x^2 - 24*x + 72 To: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field Defn: c2 |--> -6*a + 12, None) ] sage: K.subfields(8, 'w') [ (Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9, Ring morphism: From: Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9 To: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field Defn: w0 |--> (-1/2*b*a + 1/2*b + 1/2)*c, Relative number field morphism: From: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field To: Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9 Defn: c |--> -1/3*w0^7 + 4*w0^5 - 12*w0^3 + 11*w0 a |--> 1/3*w0^6 - 10/3*w0^4 + 5*w0^2 b |--> -2/3*w0^6 + 7*w0^4 - 14*w0^2 + 6) ] sage: K.subfields(3) []
- uniformizer(P, others='positive')¶
Returns an element of self with valuation 1 at the prime ideal P.
INPUT:
self
- a number fieldP
- a prime ideal of selfothers
- either “positive” (default), in which case the element will have non-negative valuation at all other primes of self, or “negative”, in which case the element will have non-positive 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: K.<a, b> = NumberField([x^2 + 23, x^2 - 3]) sage: P = K.prime_factors(5)[0]; P Fractional ideal (5, 1/2*a + b - 5/2) sage: u = K.uniformizer(P) sage: u.valuation(P) 1 sage: (P, 1) in K.factor(u) True
- vector_space(*args, **kwds)¶
For a relative number field,
vector_space()
is deliberately not implemented, so that a user cannot confuserelative_vector_space()
withabsolute_vector_space()
.EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2]) sage: K.vector_space() Traceback (most recent call last): ... NotImplementedError: For a relative number field L you must use either L.relative_vector_space() or L.absolute_vector_space() as appropriate
- sage.rings.number_field.number_field_rel.NumberField_relative_v1(base_field, poly, name, latex_name, canonical_embedding=None)¶
Used for unpickling old pickles.
EXAMPLES:
sage: from sage.rings.number_field.number_field_rel import NumberField_relative_v1 sage: R.<x> = CyclotomicField(3)[] sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a') Number Field in a with defining polynomial x^2 + 7 over its base field
- sage.rings.number_field.number_field_rel.is_RelativeNumberField(x)¶
Return
True
if \(x\) is a relative number field.EXAMPLES:
sage: from sage.rings.number_field.number_field_rel import is_RelativeNumberField sage: is_RelativeNumberField(NumberField(x^2+1,'a')) False sage: k.<a> = NumberField(x^3 - 2) sage: l.<b> = k.extension(x^3 - 3); l Number Field in b with defining polynomial x^3 - 3 over its base field sage: is_RelativeNumberField(l) True sage: is_RelativeNumberField(QQ) False