Fields¶

class sage.categories.fields.Fields(base_category)[source]¶

Bases: CategoryWithAxiom_singleton

The category of (commutative) fields, i.e. commutative rings where all nonzero elements have multiplicative inverses

EXAMPLES:

Sage

sage: K = Fields()
sage: K
Category of fields
sage: Fields().super_categories()
[Category of euclidean domains,
 Category of division rings,
 Category of noetherian rings]

sage: K(IntegerRing())
Rational Field
sage: K(PolynomialRing(GF(3), 'x'))
Fraction Field of Univariate Polynomial Ring in x over
Finite Field of size 3
sage: K(RealField())                                                            # needs sage.rings.real_mpfr
Real Field with 53 bits of precision

Python

>>> from sage.all import *
>>> K = Fields()
>>> K
Category of fields
>>> Fields().super_categories()
[Category of euclidean domains,
 Category of division rings,
 Category of noetherian rings]

>>> K(IntegerRing())
Rational Field
>>> K(PolynomialRing(GF(Integer(3)), 'x'))
Fraction Field of Univariate Polynomial Ring in x over
Finite Field of size 3
>>> K(RealField())                                                            # needs sage.rings.real_mpfr
Real Field with 53 bits of precision

class ElementMethods[source]¶

Bases: object

euclidean_degree()[source]¶

Return the degree of this element as an element of a Euclidean domain.

In a field, this returns 0 for all but the zero element (for which it is undefined).

EXAMPLES:

Sage

sage: QQ.one().euclidean_degree()
0

Python

>>> from sage.all import *
>>> QQ.one().euclidean_degree()
0

factor()[source]¶

Return a factorization of self.

Since self is either a unit or zero, this function is trivial.

EXAMPLES:

Sage

sage: x = GF(7)(5)
sage: x.factor()
5
sage: RR(0).factor()                                                    # needs sage.rings.real_mpfr
Traceback (most recent call last):
...
ArithmeticError: factorization of 0.000000000000000 is not defined

Python

>>> from sage.all import *
>>> x = GF(Integer(7))(Integer(5))
>>> x.factor()
5
>>> RR(Integer(0)).factor()                                                    # needs sage.rings.real_mpfr
Traceback (most recent call last):
...
ArithmeticError: factorization of 0.000000000000000 is not defined

gcd(other)[source]¶

Greatest common divisor.

Note

Since we are in a field and the greatest common divisor is only determined up to a unit, it is correct to either return zero or one. Note that fraction fields of unique factorization domains provide a more sophisticated gcd.

EXAMPLES:

Sage

sage: K = GF(5)
sage: K(2).gcd(K(1))
1
sage: K(0).gcd(K(0))
0
sage: all(x.gcd(y) == (0 if x == 0 and y == 0 else 1)
....:     for x in K for y in K)
True

Python

>>> from sage.all import *
>>> K = GF(Integer(5))
>>> K(Integer(2)).gcd(K(Integer(1)))
1
>>> K(Integer(0)).gcd(K(Integer(0)))
0
>>> all(x.gcd(y) == (Integer(0) if x == Integer(0) and y == Integer(0) else Integer(1))
...     for x in K for y in K)
True

For field of characteristic zero, the gcd of integers is considered as if they were elements of the integer ring:

Sage

sage: gcd(15.0,12.0)                                                    # needs sage.rings.real_mpfr
3.00000000000000

Python

>>> from sage.all import *
>>> gcd(RealNumber('15.0'),RealNumber('12.0'))                                                    # needs sage.rings.real_mpfr
3.00000000000000

But for other floating point numbers, the gcd is just \(0.0\) or \(1.0\):

Sage

sage: gcd(3.2, 2.18)                                                    # needs sage.rings.real_mpfr
1.00000000000000

sage: gcd(0.0, 0.0)                                                     # needs sage.rings.real_mpfr
0.000000000000000

Python

>>> from sage.all import *
>>> gcd(RealNumber('3.2'), RealNumber('2.18'))                                                    # needs sage.rings.real_mpfr
1.00000000000000

>>> gcd(RealNumber('0.0'), RealNumber('0.0'))                                                     # needs sage.rings.real_mpfr
0.000000000000000

AUTHOR:

Simon King (2011-02) – Issue #10771
Vincent Delecroix (2015) – Issue #17671

inverse_of_unit()[source]¶

Return the inverse of this element.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: NumberField(x^7 + 2, 'a')(2).inverse_of_unit()                    # needs sage.rings.number_field
1/2

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> NumberField(x**Integer(7) + Integer(2), 'a')(Integer(2)).inverse_of_unit()                    # needs sage.rings.number_field
1/2

Trying to invert the zero element typically raises a ZeroDivisionError:

Sage

sage: QQ(0).inverse_of_unit()
Traceback (most recent call last):
...
ZeroDivisionError: rational division by zero

Python

>>> from sage.all import *
>>> QQ(Integer(0)).inverse_of_unit()
Traceback (most recent call last):
...
ZeroDivisionError: rational division by zero

To catch that exception in a way that also works for non-units in more general rings, use something like:

Sage

sage: try:
....:    QQ(0).inverse_of_unit()
....: except ArithmeticError:
....:    pass

Python

>>> from sage.all import *
>>> try:
...    QQ(Integer(0)).inverse_of_unit()
... except ArithmeticError:
...    pass

Also note that some “fields” allow one to invert the zero element:

Sage

sage: RR(0).inverse_of_unit()
+infinity

Python

>>> from sage.all import *
>>> RR(Integer(0)).inverse_of_unit()
+infinity

is_unit()[source]¶

Return True if self has a multiplicative inverse.

EXAMPLES:

Sage

sage: QQ(2).is_unit()
True
sage: QQ(0).is_unit()
False

Python

>>> from sage.all import *
>>> QQ(Integer(2)).is_unit()
True
>>> QQ(Integer(0)).is_unit()
False

lcm(other)[source]¶

Least common multiple.

Note

Since we are in a field and the least common multiple is only determined up to a unit, it is correct to either return zero or one. Note that fraction fields of unique factorization domains provide a more sophisticated lcm.

EXAMPLES:

Sage

sage: GF(2)(1).lcm(GF(2)(0))
0
sage: GF(2)(1).lcm(GF(2)(1))
1

Python

>>> from sage.all import *
>>> GF(Integer(2))(Integer(1)).lcm(GF(Integer(2))(Integer(0)))
0
>>> GF(Integer(2))(Integer(1)).lcm(GF(Integer(2))(Integer(1)))
1

For field of characteristic zero, the lcm of integers is considered as if they were elements of the integer ring:

Sage

sage: lcm(15.0, 12.0)                                                   # needs sage.rings.real_mpfr
60.0000000000000

Python

>>> from sage.all import *
>>> lcm(RealNumber('15.0'), RealNumber('12.0'))                                                   # needs sage.rings.real_mpfr
60.0000000000000

But for others floating point numbers, it is just \(0.0\) or \(1.0\):

Sage

sage: lcm(3.2, 2.18)                                                    # needs sage.rings.real_mpfr
1.00000000000000

sage: lcm(0.0, 0.0)                                                     # needs sage.rings.real_mpfr
0.000000000000000

Python

>>> from sage.all import *
>>> lcm(RealNumber('3.2'), RealNumber('2.18'))                                                    # needs sage.rings.real_mpfr
1.00000000000000

>>> lcm(RealNumber('0.0'), RealNumber('0.0'))                                                     # needs sage.rings.real_mpfr
0.000000000000000

AUTHOR:

Simon King (2011-02) – Issue #10771
Vincent Delecroix (2015) – Issue #17671

quo_rem(other)[source]¶

Return the quotient with remainder of the division of this element by other.

INPUT:

other – an element of the field

EXAMPLES:

Sage

sage: f,g = QQ(1), QQ(2)
sage: f.quo_rem(g)
(1/2, 0)

Python

>>> from sage.all import *
>>> f,g = QQ(Integer(1)), QQ(Integer(2))
>>> f.quo_rem(g)
(1/2, 0)

xgcd(other)[source]¶

Compute the extended gcd of self and other.

INPUT:

other – an element with the same parent as self

OUTPUT:

A tuple (r, s, t) of elements in the parent of self such that r = s * self + t * other. Since the computations are done over a field, r is zero if self and other are zero, and one otherwise.

AUTHORS:

Julian Rueth (2012-10-19): moved here from sage.structure.element.FieldElement

EXAMPLES:

Sage

sage: K = GF(5)
sage: K(2).xgcd(K(1))
(1, 3, 0)
sage: K(0).xgcd(K(4))
(1, 0, 4)
sage: K(1).xgcd(K(1))
(1, 1, 0)
sage: GF(5)(0).xgcd(GF(5)(0))
(0, 0, 0)

Python

>>> from sage.all import *
>>> K = GF(Integer(5))
>>> K(Integer(2)).xgcd(K(Integer(1)))
(1, 3, 0)
>>> K(Integer(0)).xgcd(K(Integer(4)))
(1, 0, 4)
>>> K(Integer(1)).xgcd(K(Integer(1)))
(1, 1, 0)
>>> GF(Integer(5))(Integer(0)).xgcd(GF(Integer(5))(Integer(0)))
(0, 0, 0)

The xgcd of nonzero floating point numbers will be a triple of floating points. But if the input are two integral floating points the result is a floating point version of the standard gcd on \(\ZZ\):

Sage

sage: xgcd(12.0, 8.0)                                                   # needs sage.rings.real_mpfr
(4.00000000000000, 1.00000000000000, -1.00000000000000)

sage: xgcd(3.1, 2.98714)                                                # needs sage.rings.real_mpfr
(1.00000000000000, 0.322580645161290, 0.000000000000000)

sage: xgcd(0.0, 1.1)                                                    # needs sage.rings.real_mpfr
(1.00000000000000, 0.000000000000000, 0.909090909090909)

Python

>>> from sage.all import *
>>> xgcd(RealNumber('12.0'), RealNumber('8.0'))                                                   # needs sage.rings.real_mpfr
(4.00000000000000, 1.00000000000000, -1.00000000000000)

>>> xgcd(RealNumber('3.1'), RealNumber('2.98714'))                                                # needs sage.rings.real_mpfr
(1.00000000000000, 0.322580645161290, 0.000000000000000)

>>> xgcd(RealNumber('0.0'), RealNumber('1.1'))                                                    # needs sage.rings.real_mpfr
(1.00000000000000, 0.000000000000000, 0.909090909090909)

Finite[source]¶: alias of FiniteFields

class ParentMethods[source]¶

Bases: object

fraction_field()[source]¶

Return the fraction field of self, which is self.

EXAMPLES:

Sage

sage: QQ.fraction_field() is QQ
True

Python

>>> from sage.all import *
>>> QQ.fraction_field() is QQ
True

ideal(*gens, **kwds)[source]¶

Return the ideal generated by gens.

INPUT:

an element or a list/tuple/sequence of elements, the generators

Any named arguments are ignored.

EXAMPLES:

Sage

sage: QQ.ideal(2)
Principal ideal (1) of Rational Field
sage: QQ.ideal(0)
Principal ideal (0) of Rational Field

Python

>>> from sage.all import *
>>> QQ.ideal(Integer(2))
Principal ideal (1) of Rational Field
>>> QQ.ideal(Integer(0))
Principal ideal (0) of Rational Field

is_field(proof=True)[source]¶

Return True as self is a field.

EXAMPLES:

Sage

sage: QQ.is_field()
True
sage: Parent(QQ,category=Fields()).is_field()
True

Python

>>> from sage.all import *
>>> QQ.is_field()
True
>>> Parent(QQ,category=Fields()).is_field()
True

is_integrally_closed()[source]¶

Return True, as per IntegralDomain.is_integrally_closed(): for every field \(F\), \(F\) is its own field of fractions, hence every element of \(F\) is integral over \(F\).

EXAMPLES:

Sage

sage: QQ.is_integrally_closed()
True
sage: QQbar.is_integrally_closed()                                      # needs sage.rings.number_field
True
sage: Z5 = GF(5); Z5
Finite Field of size 5
sage: Z5.is_integrally_closed()
True

Python

>>> from sage.all import *
>>> QQ.is_integrally_closed()
True
>>> QQbar.is_integrally_closed()                                      # needs sage.rings.number_field
True
>>> Z5 = GF(Integer(5)); Z5
Finite Field of size 5
>>> Z5.is_integrally_closed()
True

is_perfect()[source]¶

Return whether this field is perfect, i.e., its characteristic is \(p=0\) or every element has a \(p\)-th root.

EXAMPLES:

Sage

sage: QQ.is_perfect()
True
sage: GF(2).is_perfect()
True
sage: FunctionField(GF(2), 'x').is_perfect()
False

Python

>>> from sage.all import *
>>> QQ.is_perfect()
True
>>> GF(Integer(2)).is_perfect()
True
>>> FunctionField(GF(Integer(2)), 'x').is_perfect()
False

vector_space(*args, **kwds)[source]¶

Give an isomorphism of this field with a vector space over a subfield.

This method is an alias for free_module, which may have more documentation.

INPUT:

base – a subfield or morphism into this field (defaults to the base field)
basis – a basis of the field as a vector space over the subfield; if not given, one is chosen automatically
map – whether to return maps from and to the vector space

OUTPUT:

V – a vector space over base
from_V – an isomorphism from V to this field
to_V – the inverse isomorphism from this field to V

EXAMPLES:

Sage

sage: # needs sage.rings.padics
sage: K.<a> = Qq(125)
sage: V, fr, to = K.vector_space()
sage: v = V([1, 2, 3])
sage: fr(v, 7)
(3*a^2 + 2*a + 1) + O(5^7)

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> K = Qq(Integer(125), names=('a',)); (a,) = K._first_ngens(1)
>>> V, fr, to = K.vector_space()
>>> v = V([Integer(1), Integer(2), Integer(3)])
>>> fr(v, Integer(7))
(3*a^2 + 2*a + 1) + O(5^7)

extra_super_categories()[source]¶

EXAMPLES:

Sage

sage: Fields().extra_super_categories()
[Category of euclidean domains, Category of noetherian rings]

Python

>>> from sage.all import *
>>> Fields().extra_super_categories()
[Category of euclidean domains, Category of noetherian rings]