def construction(self):
"""
Returns the functorial construction of ``self``, namely,
completion of the rational numbers with respect a given prime.
Also preserves other information that makes this field unique
(e.g. precision, rounding, print mode).
EXAMPLE::
sage: K = Qp(17, 8, print_mode='val-unit', print_sep='&')
sage: c, L = K.construction(); L
Rational Field
sage: c(L)
17-adic Field with capped relative precision 8
sage: K == c(L)
True
"""
from sage.categories.pushout import CompletionFunctor
return (CompletionFunctor(
self.prime(), self.precision_cap(), {
'print_mode': self._printer.dict(),
'type': self._prec_type(),
'names': self._names
}), QQ)
def construction(self):
r"""
Return the functorial construction of this Laurent power series ring.
The construction is given as the completion of the Laurent polynomials.
EXAMPLES::
sage: L.<t> = LaurentSeriesRing(ZZ, default_prec=42)
sage: phi, arg = L.construction()
sage: phi
Completion[t, prec=42]
sage: arg
Univariate Laurent Polynomial Ring in t over Integer Ring
sage: phi(arg) is L
True
Because of this construction, pushout is automatically available::
sage: 1/2 * t
1/2*t
sage: parent(1/2 * t)
Laurent Series Ring in t over Rational Field
sage: QQbar.gen() * t
I*t
sage: parent(QQbar.gen() * t)
Laurent Series Ring in t over Algebraic Field
"""
from sage.categories.pushout import CompletionFunctor
from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
L = LaurentPolynomialRing(self.base_ring(), self._names[0])
return CompletionFunctor(self._names[0], self.default_prec()), L
def construction(self, forbid_frac_field=False):
"""
Returns the functorial construction of ``self``, namely,
completion of the rational numbers with respect a given prime.
Also preserves other information that makes this field unique
(e.g. precision, rounding, print mode).
INPUT:
- ``forbid_frac_field`` -- require a completion functor rather
than a fraction field functor. This is used in the
:meth:`sage.rings.padics.local_generic.LocalGeneric.change` method.
EXAMPLES::
sage: K = Qp(17, 8, print_mode='val-unit', print_sep='&')
sage: c, L = K.construction(); L
17-adic Ring with capped relative precision 8
sage: c
FractionField
sage: c(L)
17-adic Field with capped relative precision 8
sage: K == c(L)
True
We can get a completion functor by forbidding the fraction field::
sage: c, L = K.construction(forbid_frac_field=True); L
Rational Field
sage: c
Completion[17, prec=8]
sage: c(L)
17-adic Field with capped relative precision 8
sage: K == c(L)
True
TESTS::
sage: R = QpLC(13,(31,41))
sage: R._precision_cap()
(31, 41)
sage: F, Z = R.construction()
sage: S = F(Z)
sage: S._precision_cap()
(31, 41)
"""
from sage.categories.pushout import FractionField, CompletionFunctor
if forbid_frac_field:
extras = {
'print_mode': self._printer.dict(),
'type': self._prec_type(),
'names': self._names
}
if hasattr(self, '_label'):
extras['label'] = self._label
return (CompletionFunctor(self.prime(), self._precision_cap(),
extras), QQ)
else:
return FractionField(), self.integer_ring()
def construction(self):
"""
Return the functorial construction of self, namely, completion of
the univariate polynomial ring with respect to the indeterminate
(to a given precision).
EXAMPLES::
sage: R = PowerSeriesRing(ZZ, 'x')
sage: c, S = R.construction(); S
Univariate Polynomial Ring in x over Integer Ring
sage: R == c(S)
True
sage: R = PowerSeriesRing(ZZ, 'x', sparse=True)
sage: c, S = R.construction()
sage: R == c(S)
True
"""
from sage.categories.pushout import CompletionFunctor
if self.is_sparse():
extras = {'sparse': True}
else:
extras = None
return CompletionFunctor(self._names[0], self.default_prec(),
extras), self._poly_ring()
def construction(self):
"""
Returns the functorial construction of ``self``, namely,
completion of the rational numbers with respect a given prime.
Also preserves other information that makes this field unique
(e.g. precision, rounding, print mode).
EXAMPLES::
sage: K = Qp(17, 8, print_mode='val-unit', print_sep='&')
sage: c, L = K.construction(); L
Rational Field
sage: c(L)
17-adic Field with capped relative precision 8
sage: K == c(L)
True
TESTS::
sage: R = QpLC(13,(31,41))
sage: R._precision_cap()
(31, 41)
sage: F, Z = R.construction()
sage: S = F(Z)
sage: S._precision_cap()
(31, 41)
"""
from sage.categories.pushout import CompletionFunctor
extras = {'print_mode':self._printer.dict(), 'type':self._prec_type(), 'names':self._names}
if hasattr(self, '_label'):
extras['label'] = self._label
return (CompletionFunctor(self.prime(), self._precision_cap(), extras), QQ)
def construction(self):
"""
Returns a functor F and base ring R such that F(R) == self.
EXAMPLES::
sage: M = PowerSeriesRing(QQ,4,'f'); M
Multivariate Power Series Ring in f0, f1, f2, f3 over Rational Field
sage: (c,R) = M.construction(); (c,R)
(Completion[('f0', 'f1', 'f2', 'f3'), prec=12],
Multivariate Polynomial Ring in f0, f1, f2, f3 over Rational Field)
sage: c
Completion[('f0', 'f1', 'f2', 'f3'), prec=12]
sage: c(R)
Multivariate Power Series Ring in f0, f1, f2, f3 over Rational Field
sage: c(R) == M
True
TESTS::
sage: M2 = PowerSeriesRing(QQ,4,'f', sparse=True)
sage: M == M2
False
sage: c,R = M2.construction()
sage: c(R)==M2
True
sage: M3 = PowerSeriesRing(QQ,4,'f', order='degrevlex')
sage: M3 == M
False
sage: M3 == M2
False
sage: c,R = M3.construction()
sage: c(R)==M3
True
"""
from sage.categories.pushout import CompletionFunctor
extras = {'order': self.term_order(), 'num_gens': self.ngens()}
if self.is_sparse():
extras['sparse'] = True
return (CompletionFunctor(self._names,
self.default_prec(),
extras=extras), self._poly_ring())
def construction(self, forbid_frac_field=False):
"""
Returns the functorial construction of self, namely,
completion of the rational numbers with respect a given prime.
Also preserves other information that makes this field unique
(e.g. precision, rounding, print mode).
INPUT:
- ``forbid_frac_field`` -- ignored, for compatibility with other p-adic types.
EXAMPLES::
sage: K = Zp(17, 8, print_mode='val-unit', print_sep='&')
sage: c, L = K.construction(); L
Integer Ring
sage: c(L)
17-adic Ring with capped relative precision 8
sage: K == c(L)
True
TESTS::
sage: R = ZpLC(13,(31,41))
sage: R._precision_cap()
(31, 41)
sage: F, Z = R.construction()
sage: S = F(Z)
sage: S._precision_cap()
(31, 41)
"""
from sage.categories.pushout import CompletionFunctor
extras = {
'print_mode': self._printer.dict(),
'type': self._prec_type(),
'names': self._names
}
if hasattr(self, '_label'):
extras['label'] = self._label
return (CompletionFunctor(self.prime(), self._precision_cap(),
extras), ZZ)
def construction(self):
"""
Returns a functor F and base ring R such that F(R) == self.
EXAMPLES::
sage: M = PowerSeriesRing(QQ,4,'f'); M
Multivariate Power Series Ring in f0, f1, f2, f3 over Rational Field
sage: (c,R) = M.construction(); (c,R)
(Completion[('f0', 'f1', 'f2', 'f3')],
Multivariate Polynomial Ring in f0, f1, f2, f3 over Rational Field)
sage: c
Completion[('f0', 'f1', 'f2', 'f3')]
sage: c(R)
Multivariate Power Series Ring in f0, f1, f2, f3 over Rational Field
sage: c(R) == M
True
"""
from sage.categories.pushout import CompletionFunctor
return (CompletionFunctor(self._names,
self.default_prec()), self._poly_ring())