Fraction Field of Integral Domains¶

AUTHORS:

William Stein (with input from David Joyner, David Kohel, and Joe Wetherell)
Burcin Erocal
Julian Rüth (2017-06-27): embedding into the field of fractions and its section

EXAMPLES:

Quotienting is a constructor for an element of the fraction field:

Sage

sage: R.<x> = QQ[]
sage: (x^2-1)/(x+1)
x - 1
sage: parent((x^2-1)/(x+1))
Fraction Field of Univariate Polynomial Ring in x over Rational Field

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> (x**Integer(2)-Integer(1))/(x+Integer(1))
x - 1
>>> parent((x**Integer(2)-Integer(1))/(x+Integer(1)))
Fraction Field of Univariate Polynomial Ring in x over Rational Field

Sage Live

R.<x> = QQ[]
(x^2-1)/(x+1)
parent((x^2-1)/(x+1))

The GCD is not taken (since it doesn’t converge sometimes) in the inexact case:

Sage

sage: # needs sage.rings.real_mpfr
sage: Z.<z> = CC[]
sage: I = CC.gen()
sage: (1+I+z)/(z+0.1*I)
(z + 1.00000000000000 + I)/(z + 0.100000000000000*I)
sage: (1+I*z)/(z+1.1)
(I*z + 1.00000000000000)/(z + 1.10000000000000)

Python

>>> from sage.all import *
>>> # needs sage.rings.real_mpfr
>>> Z = CC['z']; (z,) = Z._first_ngens(1)
>>> I = CC.gen()
>>> (Integer(1)+I+z)/(z+RealNumber('0.1')*I)
(z + 1.00000000000000 + I)/(z + 0.100000000000000*I)
>>> (Integer(1)+I*z)/(z+RealNumber('1.1'))
(I*z + 1.00000000000000)/(z + 1.10000000000000)

Sage Live

# needs sage.rings.real_mpfr
Z.<z> = CC[]
I = CC.gen()
(1+I+z)/(z+0.1*I)
(1+I*z)/(z+1.1)

sage.rings.fraction_field.FractionField(R, names=None)[source]¶

Create the fraction field of the integral domain R.

INPUT:

R – an integral domain
names – ignored

EXAMPLES:

We create some example fraction fields:

Sage

sage: FractionField(IntegerRing())
Rational Field
sage: FractionField(PolynomialRing(RationalField(),'x'))
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: FractionField(PolynomialRing(IntegerRing(),'x'))
Fraction Field of Univariate Polynomial Ring in x over Integer Ring
sage: FractionField(PolynomialRing(RationalField(),2,'x'))
Fraction Field of Multivariate Polynomial Ring in x0, x1 over Rational Field

Python

>>> from sage.all import *
>>> FractionField(IntegerRing())
Rational Field
>>> FractionField(PolynomialRing(RationalField(),'x'))
Fraction Field of Univariate Polynomial Ring in x over Rational Field
>>> FractionField(PolynomialRing(IntegerRing(),'x'))
Fraction Field of Univariate Polynomial Ring in x over Integer Ring
>>> FractionField(PolynomialRing(RationalField(),Integer(2),'x'))
Fraction Field of Multivariate Polynomial Ring in x0, x1 over Rational Field

Sage Live

FractionField(IntegerRing())
FractionField(PolynomialRing(RationalField(),'x'))
FractionField(PolynomialRing(IntegerRing(),'x'))
FractionField(PolynomialRing(RationalField(),2,'x'))

Dividing elements often implicitly creates elements of the fraction field:

Sage

sage: x = PolynomialRing(RationalField(), 'x').gen()
sage: f = x/(x+1)
sage: g = x**3/(x+1)
sage: f/g
1/x^2
sage: g/f
x^2

Python

>>> from sage.all import *
>>> x = PolynomialRing(RationalField(), 'x').gen()
>>> f = x/(x+Integer(1))
>>> g = x**Integer(3)/(x+Integer(1))
>>> f/g
1/x^2
>>> g/f
x^2

Sage Live

x = PolynomialRing(RationalField(), 'x').gen()
f = x/(x+1)
g = x**3/(x+1)
f/g
g/f

The input must be an integral domain:

Sage

sage: Frac(Integers(4))
Traceback (most recent call last):
...
TypeError: R must be an integral domain

Python

>>> from sage.all import *
>>> Frac(Integers(Integer(4)))
Traceback (most recent call last):
...
TypeError: R must be an integral domain

Sage Live

Frac(Integers(4))

class sage.rings.fraction_field.FractionFieldEmbedding[source]¶

Bases: DefaultConvertMap_unique

The embedding of an integral domain into its field of fractions.

EXAMPLES:

Sage

sage: R.<x> = QQ[]
sage: f = R.fraction_field().coerce_map_from(R); f
Coercion map:
  From: Univariate Polynomial Ring in x over Rational Field
  To:   Fraction Field of Univariate Polynomial Ring in x over Rational Field

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = R.fraction_field().coerce_map_from(R); f
Coercion map:
  From: Univariate Polynomial Ring in x over Rational Field
  To:   Fraction Field of Univariate Polynomial Ring in x over Rational Field

Sage Live

R.<x> = QQ[]
f = R.fraction_field().coerce_map_from(R); f

is_injective()[source]¶

Return whether this map is injective.

EXAMPLES:

The map from an integral domain to its fraction field is always injective:

Sage

sage: R.<x> = QQ[]
sage: R.fraction_field().coerce_map_from(R).is_injective()
True

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> R.fraction_field().coerce_map_from(R).is_injective()
True

Sage Live

R.<x> = QQ[]
R.fraction_field().coerce_map_from(R).is_injective()

is_surjective()[source]¶

Return whether this map is surjective.

EXAMPLES:

Sage

sage: R.<x> = QQ[]
sage: R.fraction_field().coerce_map_from(R).is_surjective()
False

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> R.fraction_field().coerce_map_from(R).is_surjective()
False

Sage Live

R.<x> = QQ[]
R.fraction_field().coerce_map_from(R).is_surjective()

section()[source]¶

Return a section of this map.

EXAMPLES:

Sage

sage: R.<x> = QQ[]
sage: R.fraction_field().coerce_map_from(R).section()
Section map:
  From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
  To:   Univariate Polynomial Ring in x over Rational Field

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> R.fraction_field().coerce_map_from(R).section()
Section map:
  From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
  To:   Univariate Polynomial Ring in x over Rational Field

Sage Live

R.<x> = QQ[]
R.fraction_field().coerce_map_from(R).section()

class sage.rings.fraction_field.FractionFieldEmbeddingSection[source]¶

Bases: Section

The section of the embedding of an integral domain into its field of fractions.

EXAMPLES:

Sage

sage: R.<x> = QQ[]
sage: f = R.fraction_field().coerce_map_from(R).section(); f
Section map:
  From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
  To:   Univariate Polynomial Ring in x over Rational Field

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = R.fraction_field().coerce_map_from(R).section(); f
Section map:
  From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
  To:   Univariate Polynomial Ring in x over Rational Field

Sage Live

R.<x> = QQ[]
f = R.fraction_field().coerce_map_from(R).section(); f

class sage.rings.fraction_field.FractionField_1poly_field(R, element_class=<class 'sage.rings.fraction_field_element.FractionFieldElement_1poly_field'>)[source]¶

Bases: FractionField_generic

The fraction field of a univariate polynomial ring over a field.

Many of the functions here are included for coherence with number fields.

class_number()[source]¶

Here for compatibility with number fields and function fields.

EXAMPLES:

Sage

sage: R.<t> = GF(5)[]; K = R.fraction_field()
sage: K.class_number()
1

Python

>>> from sage.all import *
>>> R = GF(Integer(5))['t']; (t,) = R._first_ngens(1); K = R.fraction_field()
>>> K.class_number()
1

Sage Live

R.<t> = GF(5)[]; K = R.fraction_field()
K.class_number()

function_field()[source]¶

Return the isomorphic function field.

EXAMPLES:

Sage

sage: R.<t> = GF(5)[]
sage: K = R.fraction_field()
sage: K.function_field()
Rational function field in t over Finite Field of size 5

Python

>>> from sage.all import *
>>> R = GF(Integer(5))['t']; (t,) = R._first_ngens(1)
>>> K = R.fraction_field()
>>> K.function_field()
Rational function field in t over Finite Field of size 5

Sage Live

R.<t> = GF(5)[]
K = R.fraction_field()
K.function_field()

See also

sage.rings.function_field.RationalFunctionField.field()

maximal_order()[source]¶

Return the maximal order in this fraction field.

EXAMPLES:

Sage

sage: K = FractionField(GF(5)['t'])
sage: K.maximal_order()
Univariate Polynomial Ring in t over Finite Field of size 5

Python

>>> from sage.all import *
>>> K = FractionField(GF(Integer(5))['t'])
>>> K.maximal_order()
Univariate Polynomial Ring in t over Finite Field of size 5

Sage Live

K = FractionField(GF(5)['t'])
K.maximal_order()

ring_of_integers()[source]¶

Return the ring of integers in this fraction field.

EXAMPLES:

Sage

sage: K = FractionField(GF(5)['t'])
sage: K.ring_of_integers()
Univariate Polynomial Ring in t over Finite Field of size 5

Python

>>> from sage.all import *
>>> K = FractionField(GF(Integer(5))['t'])
>>> K.ring_of_integers()
Univariate Polynomial Ring in t over Finite Field of size 5

Sage Live

K = FractionField(GF(5)['t'])
K.ring_of_integers()

class sage.rings.fraction_field.FractionField_generic(R, element_class=<class 'sage.rings.fraction_field_element.FractionFieldElement'>, category=Category of quotient fields)[source]¶

Bases: Field

The fraction field of an integral domain.

base_ring()[source]¶

Return the base ring of self.

This is the base ring of the ring which this fraction field is the fraction field of.

EXAMPLES:

Sage

sage: R = Frac(ZZ['t'])
sage: R.base_ring()
Integer Ring

Python

>>> from sage.all import *
>>> R = Frac(ZZ['t'])
>>> R.base_ring()
Integer Ring

Sage Live

R = Frac(ZZ['t'])
R.base_ring()

characteristic()[source]¶

Return the characteristic of this fraction field.

EXAMPLES:

Sage

sage: R = Frac(ZZ['t'])
sage: R.base_ring()
Integer Ring
sage: R = Frac(ZZ['t']); R.characteristic()
0
sage: R = Frac(GF(5)['w']); R.characteristic()
5

Python

>>> from sage.all import *
>>> R = Frac(ZZ['t'])
>>> R.base_ring()
Integer Ring
>>> R = Frac(ZZ['t']); R.characteristic()
0
>>> R = Frac(GF(Integer(5))['w']); R.characteristic()
5

Sage Live

R = Frac(ZZ['t'])
R.base_ring()
R = Frac(ZZ['t']); R.characteristic()
R = Frac(GF(5)['w']); R.characteristic()

construction()[source]¶

EXAMPLES:

Sage

sage: Frac(ZZ['x']).construction()
(FractionField, Univariate Polynomial Ring in x over Integer Ring)
sage: K = Frac(GF(3)['t'])
sage: f, R = K.construction()
sage: f(R)
Fraction Field of Univariate Polynomial Ring in t
 over Finite Field of size 3
sage: f(R) == K
True

Python

>>> from sage.all import *
>>> Frac(ZZ['x']).construction()
(FractionField, Univariate Polynomial Ring in x over Integer Ring)
>>> K = Frac(GF(Integer(3))['t'])
>>> f, R = K.construction()
>>> f(R)
Fraction Field of Univariate Polynomial Ring in t
 over Finite Field of size 3
>>> f(R) == K
True

Sage Live

Frac(ZZ['x']).construction()
K = Frac(GF(3)['t'])
f, R = K.construction()
f(R)
f(R) == K

gen(i=0)[source]¶

Return the i-th generator of self.

EXAMPLES:

Sage

sage: R = Frac(PolynomialRing(QQ,'z',10)); R
Fraction Field of Multivariate Polynomial Ring
 in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9 over Rational Field
sage: R.0
z0
sage: R.gen(3)
z3
sage: R.3
z3

Python

>>> from sage.all import *
>>> R = Frac(PolynomialRing(QQ,'z',Integer(10))); R
Fraction Field of Multivariate Polynomial Ring
 in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9 over Rational Field
>>> R.gen(0)
z0
>>> R.gen(Integer(3))
z3
>>> R.gen(3)
z3

Sage Live

R = Frac(PolynomialRing(QQ,'z',10)); R
R.0
R.gen(3)
R.3

is_exact()[source]¶

Return if self is exact which is if the underlying ring is exact.

EXAMPLES:

Sage

sage: Frac(ZZ['x']).is_exact()
True
sage: Frac(CDF['x']).is_exact()                                             # needs sage.rings.complex_double
False

Python

>>> from sage.all import *
>>> Frac(ZZ['x']).is_exact()
True
>>> Frac(CDF['x']).is_exact()                                             # needs sage.rings.complex_double
False

Sage Live

Frac(ZZ['x']).is_exact()
Frac(CDF['x']).is_exact()                                             # needs sage.rings.complex_double

is_field(proof=True)[source]¶

Return True, since the fraction field is a field.

EXAMPLES:

Sage

sage: Frac(ZZ).is_field()
True

Python

>>> from sage.all import *
>>> Frac(ZZ).is_field()
True

Sage Live

Frac(ZZ).is_field()

is_finite()[source]¶

Tells whether this fraction field is finite.

Note

A fraction field is finite if and only if the associated integral domain is finite.

EXAMPLES:

Sage

sage: Frac(QQ['a','b','c']).is_finite()
False

Python

>>> from sage.all import *
>>> Frac(QQ['a','b','c']).is_finite()
False

Sage Live

Frac(QQ['a','b','c']).is_finite()

ngens()[source]¶

This is the same as for the parent object.

EXAMPLES:

Sage

sage: R = Frac(PolynomialRing(QQ,'z',10)); R
Fraction Field of Multivariate Polynomial Ring
 in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9 over Rational Field
sage: R.ngens()
10

Python

>>> from sage.all import *
>>> R = Frac(PolynomialRing(QQ,'z',Integer(10))); R
Fraction Field of Multivariate Polynomial Ring
 in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9 over Rational Field
>>> R.ngens()
10

Sage Live

R = Frac(PolynomialRing(QQ,'z',10)); R
R.ngens()

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

Return a random element in this fraction field.

The arguments are passed to the random generator of the underlying ring.

EXAMPLES:

Sage

sage: F = ZZ['x'].fraction_field()
sage: F.random_element()  # random
(2*x - 8)/(-x^2 + x)

Python

>>> from sage.all import *
>>> F = ZZ['x'].fraction_field()
>>> F.random_element()  # random
(2*x - 8)/(-x^2 + x)

Sage Live

F = ZZ['x'].fraction_field()
F.random_element()  # random

Sage

sage: f = F.random_element(degree=5)
sage: f.numerator().degree() == f.denominator().degree()
True
sage: f.denominator().degree() <= 5
True
sage: while f.numerator().degree() != 5:
....:      f = F.random_element(degree=5)

Python

>>> from sage.all import *
>>> f = F.random_element(degree=Integer(5))
>>> f.numerator().degree() == f.denominator().degree()
True
>>> f.denominator().degree() <= Integer(5)
True
>>> while f.numerator().degree() != Integer(5):
...      f = F.random_element(degree=Integer(5))

Sage Live

f = F.random_element(degree=5)
f.numerator().degree() == f.denominator().degree()
f.denominator().degree() <= 5
while f.numerator().degree() != 5:
     f = F.random_element(degree=5)

Python

>>> from sage.all import *
>>> f = F.random_element(degree=Integer(5))
>>> f.numerator().degree() == f.denominator().degree()
True
>>> f.denominator().degree() <= Integer(5)
True
>>> while f.numerator().degree() != Integer(5):
...      f = F.random_element(degree=Integer(5))

Sage Live

f = F.random_element(degree=5)
f.numerator().degree() == f.denominator().degree()
f.denominator().degree() <= 5
while f.numerator().degree() != 5:
     f = F.random_element(degree=5)

ring()[source]¶

Return the ring that this is the fraction field of.

EXAMPLES:

Sage

sage: R = Frac(QQ['x,y'])
sage: R
Fraction Field of Multivariate Polynomial Ring in x, y over Rational Field
sage: R.ring()
Multivariate Polynomial Ring in x, y over Rational Field

Python

>>> from sage.all import *
>>> R = Frac(QQ['x,y'])
>>> R
Fraction Field of Multivariate Polynomial Ring in x, y over Rational Field
>>> R.ring()
Multivariate Polynomial Ring in x, y over Rational Field

Sage Live

R = Frac(QQ['x,y'])
R
R.ring()

some_elements()[source]¶

Return some elements in this field.

EXAMPLES:

Sage

sage: R.<x> = QQ[]
sage: R.fraction_field().some_elements()
[0,
 1,
 x,
 2*x,
 x/(x^2 + 2*x + 1),
 1/x^2,
 ...
 (2*x^2 + 2)/(x^2 + 2*x + 1),
 (2*x^2 + 2)/x^3,
 (2*x^2 + 2)/(x^2 - 1),
 2]

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> R.fraction_field().some_elements()
[0,
 1,
 x,
 2*x,
 x/(x^2 + 2*x + 1),
 1/x^2,
 ...
 (2*x^2 + 2)/(x^2 + 2*x + 1),
 (2*x^2 + 2)/x^3,
 (2*x^2 + 2)/(x^2 - 1),
 2]

Sage Live

R.<x> = QQ[]
R.fraction_field().some_elements()

sage.rings.fraction_field.is_FractionField(x)[source]¶

Test whether or not x inherits from FractionField_generic.

EXAMPLES:

Sage

sage: from sage.rings.fraction_field import is_FractionField
sage: is_FractionField(Frac(ZZ['x']))
doctest:warning...
DeprecationWarning: The function is_FractionField is deprecated;
use 'isinstance(..., FractionField_generic)' instead.
See https://github.com/sagemath/sage/issues/38128 for details.
True
sage: is_FractionField(QQ)
False

Python

>>> from sage.all import *
>>> from sage.rings.fraction_field import is_FractionField
>>> is_FractionField(Frac(ZZ['x']))
doctest:warning...
DeprecationWarning: The function is_FractionField is deprecated;
use 'isinstance(..., FractionField_generic)' instead.
See https://github.com/sagemath/sage/issues/38128 for details.
True
>>> is_FractionField(QQ)
False

Sage Live

from sage.rings.fraction_field import is_FractionField
is_FractionField(Frac(ZZ['x']))
is_FractionField(QQ)