def _element_constructor_(self, x, check=True):
"""
Construct an element of ``self`` from ``x``.
EXAMPLES::
sage: H = Hom(QuadraticField(-1, 'a'), QuadraticField(-1, 'b'))
sage: phi = H([H.domain().gen()]); phi
Ring morphism:
From: Number Field in a with defining polynomial x^2 + 1 with a = 1*I
To: Number Field in b with defining polynomial x^2 + 1 with b = 1*I
Defn: a |--> b
sage: H1 = End(QuadraticField(-1, 'a'))
sage: H1.coerce(loads(dumps(H1[1])))
Ring endomorphism of Number Field in a with defining polynomial x^2 + 1 with a = 1*I
Defn: a |--> -a
TESTS:
We can move morphisms between categories::
sage: f = H1.an_element()
sage: g = End(H1.domain(), category=Rings())(f)
sage: f == End(H1.domain(), category=NumberFields())(g)
True
Check that :trac:`28869` is fixed::
sage: K.<a> = CyclotomicField(8)
sage: L.<b> = K.absolute_field()
sage: H = L.Hom(K)
sage: phi = L.structure()[0]
sage: phi.parent() is H
True
sage: H(phi)
Isomorphism given by variable name change map:
From: Number Field in b with defining polynomial x^4 + 1
To: Cyclotomic Field of order 8 and degree 4
sage: R.<x> = L[]
sage: (x^2 + b).change_ring(phi)
x^2 + a
"""
if not isinstance(x, NumberFieldHomomorphism_im_gens):
return self.element_class(self, x, check=check)
from sage.categories.all import NumberFields, Rings
if (x.parent() == self or
(x.domain() == self.domain() and x.codomain() == self.codomain()
and
# This would be the better check, however it returns False currently:
# self.homset_category().is_full_subcategory(x.category_for())
# So we check instead that this is a morphism anywhere between
# Rings and NumberFields where the hom spaces do not change.
NumberFields().is_subcategory(self.homset_category())
and self.homset_category().is_subcategory(Rings())
and NumberFields().is_subcategory(x.category_for())
and x.category_for().is_subcategory(Rings()))):
return self.element_class(self, x.im_gens(), check=False)
def _normalize_number_field_data(self, R):
r"""
Helper method which returns the defining data of the number field
``R``.
EXAMPLES::
sage: sys.path.append(os.getcwd()); from mac_lane import * # optional: standalone
sage: R.<x> = QQ[]
sage: K = R.quo(x^2 + 1)
sage: pAdicValuation._normalize_number_field_data(K)
(Rational Field,
Univariate Quotient Polynomial Ring in xbar over Rational Field with modulus x^2 + 1,
x^2 + 1)
"""
from sage.rings.polynomial.polynomial_quotient_ring import is_PolynomialQuotientRing
from sage.rings.number_field.number_field import is_NumberField
from sage.rings.fraction_field import is_FractionField
if is_NumberField(R.fraction_field()):
L = R.fraction_field()
G = L.relative_polynomial()
K = L.base_ring()
elif is_PolynomialQuotientRing(R):
from sage.categories.all import NumberFields
if R.base_ring().fraction_field() not in NumberFields():
raise NotImplementedError("can not normalize quotients over %r"%(R.base_ring(),))
L = R.fraction_field()
K = R.base_ring().fraction_field()
G = R.modulus().change_ring(K)
else:
raise NotImplementedError("can not normalize %r"%(R,))
return K, L, G
def _coerce_impl(self, x):
r"""
Canonical coercion of ``x`` into this homset. The only things that
coerce canonically into self are elements of self and of homsets equal
to self.
EXAMPLES::
sage: H1 = End(QuadraticField(-1, 'a'))
sage: H1.coerce(loads(dumps(H1[1]))) # indirect doctest
Ring endomorphism of Number Field in a with defining polynomial x^2 + 1
Defn: a |--> -a
TESTS:
We can move morphisms between categories::
sage: f = H1.an_element()
sage: g = End(H1.domain(), category=Rings())(f)
sage: f == End(H1.domain(), category=NumberFields())(g)
True
"""
if not isinstance(x, NumberFieldHomomorphism_im_gens):
raise TypeError
if x.parent() is self:
return x
from sage.categories.all import NumberFields, Rings
if (x.parent() == self or
(x.domain() == self.domain() and x.codomain() == self.codomain()
and
# This would be the better check, however it returns False currently:
# self.homset_category().is_full_subcategory(x.category_for())
# So we check instead that this is a morphism anywhere between
# Rings and NumberFields where the hom spaces do not change.
NumberFields().is_subcategory(self.homset_category())
and self.homset_category().is_subcategory(Rings())
and NumberFields().is_subcategory(x.category_for())
and x.category_for().is_subcategory(Rings()))):
return NumberFieldHomomorphism_im_gens(self,
x.im_gens(),
check=False)
raise TypeError
def __init__(self, R, S, category=None):
"""
TESTS:
Check that :trac:`23647` is fixed::
sage: K.<a, b> = NumberField([x^2 - 2, x^2 - 3])
sage: e, u, v, w = End(K)
sage: e.abs_hom().parent().category()
Category of homsets of number fields
sage: (v*v).abs_hom().parent().category()
Category of homsets of number fields
"""
if category is None:
from sage.categories.all import Fields, NumberFields
if S in NumberFields():
category = NumberFields()
elif S in Fields():
category = Fields()
RingHomset_generic.__init__(self, R, S, category)