def _coerce_map_from_(self, R):
r"""
Return a coerce map from ``R`` to this field.
EXAMPLES::
sage: R.<t> = GF(5)[]
sage: K = R.fraction_field()
sage: L = K.function_field()
sage: f = K.coerce_map_from(L); f # indirect doctest
Isomorphism:
From: Rational function field in t over Finite Field of size 5
To: Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 5
sage: f(~L.gen())
1/t
"""
from sage.rings.function_field.function_field import RationalFunctionField
if isinstance(R, RationalFunctionField) and self.variable_name(
) == R.variable_name() and self.base_ring() is R.constant_base_field():
from sage.categories.all import Hom
parent = Hom(R, self)
from sage.rings.function_field.maps import FunctionFieldToFractionField
return parent.__make_element_class__(FunctionFieldToFractionField)(
parent)
return super(FractionField_1poly_field, self)._coerce_map_from_(R)
def section(self):
r"""
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
"""
from sage.categories.sets_with_partial_maps import SetsWithPartialMaps
from sage.all import Hom
parent = Hom(self.codomain(), self.domain(), SetsWithPartialMaps())
return parent.__make_element_class__(FractionFieldEmbeddingSection)(self)
def _coerce_map_from_(self, S):
"""
Return ``True`` if elements of ``S`` can be coerced into this
fraction field.
This fraction field has coercions from:
- itself
- any fraction field where the base ring coerces to the base
ring of this fraction field
- any ring that coerces to the base ring of this fraction field
EXAMPLES::
sage: F = QQ['x,y'].fraction_field()
sage: F.has_coerce_map_from(F) # indirect doctest
True
::
sage: F.has_coerce_map_from(ZZ['x,y'].fraction_field())
True
::
sage: F.has_coerce_map_from(ZZ['x,y,z'].fraction_field())
False
::
sage: F.has_coerce_map_from(ZZ)
True
Test coercions::
sage: F.coerce(1)
1
sage: F.coerce(int(1))
1
sage: F.coerce(1/2)
1/2
::
sage: K = ZZ['x,y'].fraction_field()
sage: x,y = K.gens()
sage: F.coerce(F.gen())
x
sage: F.coerce(x)
x
sage: F.coerce(x/y)
x/y
sage: L = ZZ['x'].fraction_field()
sage: K.coerce(L.gen())
x
We demonstrate that :trac:`7958` is resolved in the case of
number fields::
sage: _.<x> = ZZ[]
sage: K.<a> = NumberField(x^5-3*x^4+2424*x^3+2*x-232)
sage: R = K.ring_of_integers()
sage: S.<y> = R[]
sage: F = FractionField(S)
sage: F(1/a)
(a^4 - 3*a^3 + 2424*a^2 + 2)/232
Some corner cases have been known to fail in the past (:trac:`5917`)::
sage: F1 = FractionField( QQ['a'] )
sage: R12 = F1['x','y']
sage: R12('a')
a
sage: F1(R12(F1('a')))
a
sage: F2 = FractionField( QQ['a','b'] )
sage: R22 = F2['x','y']
sage: R22('a')
a
sage: F2(R22(F2('a')))
a
Coercion from Laurent polynomials now works (:trac:`15345`)::
sage: R = LaurentPolynomialRing(ZZ, 'x')
sage: T = PolynomialRing(ZZ, 'x')
sage: R.gen() + FractionField(T).gen()
2*x
sage: 1/(R.gen() + 1)
1/(x + 1)
sage: R = LaurentPolynomialRing(ZZ, 'x,y')
sage: FF = FractionField(PolynomialRing(ZZ, 'x,y'))
sage: prod(R.gens()) + prod(FF.gens())
2*x*y
sage: 1/(R.gen(0) + R.gen(1))
1/(x + y)
Coercion from a localization::
sage: R.<x> = ZZ[]
sage: L = Localization(R, (x**2 + 1,7))
sage: F = L.fraction_field()
sage: f = F.coerce_map_from(L); f
Coercion map:
From: Univariate Polynomial Ring in x over Integer Ring localized at (7, x^2 + 1)
To: Fraction Field of Univariate Polynomial Ring in x over Integer Ring
sage: f(L(1/7)) == 1/7
True
"""
from sage.rings.rational_field import QQ
from sage.rings.number_field.number_field_base import NumberField
from sage.rings.polynomial.laurent_polynomial_ring import \
LaurentPolynomialRing_generic
if S is self._R:
parent = self._R.Hom(self)
return parent.__make_element_class__(FractionFieldEmbedding)(
self._R, self, category=parent.homset_category())
def wrapper(x):
return self._element_class(self, x.numerator(), x.denominator())
# The case ``S`` being `\QQ` requires special handling since `\QQ` is
# not implemented as a ``FractionField_generic``.
if S is QQ and self._R.has_coerce_map_from(ZZ):
return CallableConvertMap(S,
self,
wrapper,
parent_as_first_arg=False)
# special treatment for localizations
from sage.rings.localization import Localization
if isinstance(S, Localization):
parent = S.Hom(self)
return parent.__make_element_class__(FractionFieldEmbedding)(
S, self, category=parent.homset_category())
# Number fields also need to be handled separately.
if isinstance(S, NumberField):
return CallableConvertMap(
S,
self,
self._number_field_to_frac_of_ring_of_integers,
parent_as_first_arg=False)
# special treatment for LaurentPolynomialRings
if isinstance(S, LaurentPolynomialRing_generic):
def converter(x, y=None):
if y is None:
return self._element_class(self, *x._fraction_pair())
xnum, xden = x._fraction_pair()
ynum, yden = y._fraction_pair()
return self._element_class(self, xnum * yden, xden * ynum)
return CallableConvertMap(S,
self,
converter,
parent_as_first_arg=False)
if (isinstance(S, FractionField_generic)
and self._R.has_coerce_map_from(S.ring())):
return CallableConvertMap(S,
self,
wrapper,
parent_as_first_arg=False)
if self._R.has_coerce_map_from(S):
return CallableConvertMap(S,
self,
self._element_class,
parent_as_first_arg=True)
return None