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: 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

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

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)
sage.rings.fraction_field.FractionField(R, names=None)

Create the fraction field of the integral domain R.

INPUT:

• R – an integral domain

• names – ignored

EXAMPLES:

We create some example fraction fields:

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

Dividing elements often implicitly creates elements of the fraction field:

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

The input must be an integral domain:

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

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

EXAMPLES:

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
is_injective()

Return whether this map is injective.

EXAMPLES:

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

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

Return whether this map is surjective.

EXAMPLES:

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

Return a section of this map.

EXAMPLES:

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
class sage.rings.fraction_field.FractionFieldEmbeddingSection

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

EXAMPLES:

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

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()

Here for compatibility with number fields and function fields.

EXAMPLES:

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

Return the isomorphic function field.

EXAMPLES:

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

sage.rings.function_field.RationalFunctionField.field()

maximal_order()

Return the maximal order in this fraction field.

EXAMPLES:

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

Return the ring of integers in this fraction field.

EXAMPLES:

sage: K = FractionField(GF(5)['t'])
sage: K.ring_of_integers()
Univariate Polynomial Ring in t over Finite Field of size 5
class sage.rings.fraction_field.FractionField_generic(R, element_class=<class 'sage.rings.fraction_field_element.FractionFieldElement'>, category=Category of quotient fields)

Bases: sage.rings.ring.Field

The fraction field of an integral domain.

base_ring()

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

Return the characteristic of this fraction field.

EXAMPLES:

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
construction()

EXAMPLES:

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
gen(i=0)

Return the i-th generator of self.

EXAMPLES:

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
is_exact()

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

EXAMPLES:

sage: Frac(ZZ['x']).is_exact()
True
sage: Frac(CDF['x']).is_exact()
False
is_field(proof=True)

Return True, since the fraction field is a field.

EXAMPLES:

sage: Frac(ZZ).is_field()
True
is_finite()

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

This is the same as for the parent object.

EXAMPLES:

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
random_element(*args, **kwds)

Return a random element in this fraction field.

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

EXAMPLES:

sage: F = ZZ['x'].fraction_field()
sage: F.random_element()  # random
(2*x - 8)/(-x^2 + x)
sage: f = F.random_element(degree=5)
sage: f.numerator().degree()
5
sage: f.denominator().degree()
5
ring()

Return the ring that this is the fraction field of.

EXAMPLES:

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
some_elements()

Return some elements in this field.

EXAMPLES:

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]
sage.rings.fraction_field.is_FractionField(x)

Test whether or not x inherits from FractionField_generic.

EXAMPLES:

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