def _create_(k, R):
"""
Initialization. We have to implement this as a static method
in order to call ``__make_element_class__``.
INPUT:
- ``k`` -- the residue field of ``R``, or a residue ring of ``R``.
- ``R`` -- a `p`-adic ring or field.
EXAMPLES::
sage: f = Zp(3).convert_map_from(Zmod(81))
sage: TestSuite(f).run()
"""
from sage.categories.sets_cat import Sets
from sage.categories.homset import Hom
kfield = R.residue_field()
N = k.cardinality()
q = kfield.cardinality()
n = N.exact_log(q)
if N != q**n:
raise RuntimeError("N must be a power of q")
H = Hom(k, R, Sets())
f = H.__make_element_class__(ResidueLiftingMap)(H)
f._n = n
return f
def _create_(k, R):
"""
Initialization. We have to implement this as a static method
in order to call ``__make_element_class__``.
INPUT:
- ``k`` -- the residue field of ``R``, or a residue ring of ``R``.
- ``R`` -- a `p`-adic ring or field.
EXAMPLES::
sage: f = Zp(3).convert_map_from(Zmod(81))
sage: TestSuite(f).run()
"""
from sage.categories.sets_cat import Sets
from sage.categories.homset import Hom
kfield = R.residue_field()
N = k.cardinality()
q = kfield.cardinality()
n = N.exact_log(q)
if N != q**n:
raise RuntimeError("N must be a power of q")
H = Hom(k, R, Sets())
f = H.__make_element_class__(ResidueLiftingMap)(H)
f._n = n
return f
def _convert_map_from_(self, R):
"""
Finds conversion maps from R to this ring.
Currently, a conversion exists if the defining polynomial is the same.
EXAMPLES::
sage: R.<a> = Zq(125)
sage: S = R.change(type='capped-abs', prec=40, print_mode='terse', print_pos=False)
sage: S(a - 15)
-15 + a + O(5^20)
We get conversions from the exact field::
sage: K = R.exact_field(); K
Number Field in a with defining polynomial x^3 + 3*x + 3
sage: R(K.gen())
a + O(5^20)
and its maximal order::
sage: OK = K.maximal_order()
sage: R(OK.gen(1))
a + O(5^20)
"""
cat = None
if self._implementation == 'NTL' and R == QQ:
# Want to use DefaultConvertMap_unique
return None
if isinstance(R, pAdicExtensionGeneric) and R.prime() == self.prime(
) and R.defining_polynomial(exact=True) == self.defining_polynomial(
exact=True):
if R.is_field() and not self.is_field():
cat = SetsWithPartialMaps()
elif R.category() is self.category():
cat = R.category()
else:
cat = EuclideanDomains() & MetricSpaces().Complete()
elif isinstance(R, Order) and R.number_field().defining_polynomial(
) == self.defining_polynomial():
cat = IntegralDomains()
elif isinstance(R, NumberField) and R.defining_polynomial(
) == self.defining_polynomial():
if self.is_field():
cat = Fields()
else:
cat = SetsWithPartialMaps()
else:
k = self.residue_field()
if R is k:
return ResidueLiftingMap._create_(R, self)
if cat is not None:
H = Hom(R, self, cat)
return H.__make_element_class__(DefPolyConversion)(H)
def _convert_map_from_(self, R):
"""
Finds conversion maps from R to this ring.
Currently, a conversion exists if the defining polynomial is the same.
EXAMPLES::
sage: R.<a> = Zq(125)
sage: S = R.change(type='capped-abs', prec=40, print_mode='terse', print_pos=False)
sage: S(a - 15)
-15 + a + O(5^20)
We get conversions from the exact field::
sage: K = R.exact_field(); K
Number Field in a with defining polynomial x^3 + 3*x + 3
sage: R(K.gen())
a + O(5^20)
and its maximal order::
sage: OK = K.maximal_order()
sage: R(OK.gen(1))
a + O(5^20)
"""
cat = None
if self._implementation == 'NTL' and R == QQ:
# Want to use DefaultConvertMap
return None
if isinstance(R, pAdicExtensionGeneric) and R.defining_polynomial(exact=True) == self.defining_polynomial(exact=True):
if R.is_field() and not self.is_field():
cat = SetsWithPartialMaps()
else:
cat = R.category()
elif isinstance(R, Order) and R.number_field().defining_polynomial() == self.defining_polynomial():
cat = IntegralDomains()
elif isinstance(R, NumberField) and R.defining_polynomial() == self.defining_polynomial():
if self.is_field():
cat = Fields()
else:
cat = SetsWithPartialMaps()
else:
k = self.residue_field()
if R is k:
return ResidueLiftingMap._create_(R, self)
if cat is not None:
H = Hom(R, self, cat)
return H.__make_element_class__(DefPolyConversion)(H)
def section(self):
r"""
Return the inverse map of this isomorphism.
EXAMPLES::
sage: K = QQ['x'].fraction_field()
sage: L = K.function_field()
sage: f = L.coerce_map_from(K)
sage: f.section()
Isomorphism:
From: Rational function field in x over Rational Field
To: Fraction Field of Univariate Polynomial Ring in x over Rational Field
"""
from sage.categories.all import Hom
parent = Hom(self.codomain(), self.domain())
return parent.__make_element_class__(FunctionFieldToFractionField)(parent)
def _create_(R, k):
"""
Initialization. We have to implement this as a static method
in order to call ``__make_element_class__``.
INPUT:
- ``R`` -- a `p`-adic ring or field.
- ``k`` -- the residue field of ``R``, or a residue ring of ``R``.
EXAMPLES::
sage: f = Zmod(49).convert_map_from(Zp(7))
sage: TestSuite(f).run()
sage: K.<a> = Qq(125); k = K.residue_field(); f = k.convert_map_from(K)
sage: TestSuite(f).run()
"""
if R.is_field():
from sage.categories.sets_with_partial_maps import SetsWithPartialMaps
cat = SetsWithPartialMaps()
else:
from sage.categories.rings import Rings
cat = Rings()
from sage.categories.homset import Hom
kfield = R.residue_field()
N = k.cardinality()
q = kfield.cardinality()
n = N.exact_log(q)
if N != q**n:
raise RuntimeError("N must be a power of q")
H = Hom(R, k, cat)
f = H.__make_element_class__(ResidueReductionMap)(H)
f._n = n
if kfield is k:
f._field = True
else:
f._field = False
return f
def _create_(R, k):
"""
Initialization. We have to implement this as a static method
in order to call ``__make_element_class__``.
INPUT:
- ``R`` -- a `p`-adic ring or field.
- ``k`` -- the residue field of ``R``, or a residue ring of ``R``.
EXAMPLES::
sage: f = Zmod(49).convert_map_from(Zp(7))
sage: TestSuite(f).run()
sage: K.<a> = Qq(125); k = K.residue_field(); f = k.convert_map_from(K)
sage: TestSuite(f).run()
"""
if R.is_field():
from sage.categories.sets_with_partial_maps import SetsWithPartialMaps
cat = SetsWithPartialMaps()
else:
from sage.categories.rings import Rings
cat = Rings()
from sage.categories.homset import Hom
kfield = R.residue_field()
N = k.cardinality()
q = kfield.cardinality()
n = N.exact_log(q)
if N != q**n:
raise RuntimeError("N must be a power of q")
H = Hom(R, k, cat)
f = H.__make_element_class__(ResidueReductionMap)(H)
f._n = n
if kfield is k:
f._field = True
else:
f._field = False
return f
def free_module(self, base=None, basis=None, map=True):
"""
Return a free module `V` over a specified base ring together with maps to and from `V`.
INPUT:
- ``base`` -- a subring `R` so that this ring/field is isomorphic
to a finite-rank free `R`-module `V`
- ``basis`` -- a basis for this ring/field over the base
- ``map`` -- boolean (default ``True``), whether to return
`R`-linear maps to and from `V`
OUTPUT:
- A finite-rank free `R`-module `V`
- An `R`-module isomorphism from `V` to this ring/field
(only included if ``map`` is ``True``)
- An `R`-module isomorphism from this ring/field to `V`
(only included if ``map`` is ``True``)
EXAMPLES::
sage: R.<x> = ZZ[]
sage: K.<a> = Qq(125)
sage: L.<pi> = K.extension(x^2-5)
sage: V, from_V, to_V = K.free_module()
sage: W, from_W, to_W = L.free_module()
sage: W0, from_W0, to_W0 = L.free_module(base=Qp(5))
sage: to_V(a + O(5^7))
(O(5^7), 1 + O(5^7), O(5^7))
sage: to_W(a)
(a + O(5^20), O(5^20))
sage: to_W0(a + O(5^7))
(O(5^7), 1 + O(5^7), O(5^7), O(5^7), O(5^7), O(5^7))
sage: to_W(pi)
(O(5^21), 1 + O(5^20))
sage: to_W0(pi + O(pi^11))
(O(5^6), O(5^6), O(5^6), 1 + O(5^5), O(5^5), O(5^5))
sage: X, from_X, to_X = K.free_module(K)
sage: to_X(a)
(a + O(5^20))
"""
if basis is not None:
raise NotImplementedError
B = self.base_ring()
if base is None:
base = B
A = B.base_ring()
d = self.relative_degree()
if base is B:
# May eventually want to take advantage of the fact that precision is flat
V = B**d
from_V = MapFreeModuleToOneStep
to_V = MapOneStepToFreeModule
elif base is A:
d *= B.relative_degree()
V = A**d
from_V = MapFreeModuleToTwoStep
to_V = MapTwoStepToFreeModule
elif base is self:
return super(pAdicExtensionGeneric, self).free_module(base=base,
basis=basis,
map=map)
else:
raise NotImplementedError
FromV = Hom(V, self)
ToV = Hom(self, V)
from_V = FromV.__make_element_class__(from_V)(FromV)
to_V = ToV.__make_element_class__(to_V)(ToV)
return V, from_V, to_V
