def 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.
"""
if not ring.is_Ring(R):
raise TypeError, "R must be a ring"
if not R.is_integral_domain():
raise TypeError, "R must be an integral domain."
return R.fraction_field()
def 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.
"""
if not ring.is_Ring(R):
raise TypeError, "R must be a ring"
if not R.is_integral_domain():
raise TypeError, "R must be an integral domain."
return R.fraction_field()
