def __init__(self, base, prec, names, element_class, category=None):
"""
Initializes self.
EXAMPLES::
sage: R = Zp(5) #indirect doctest
sage: R.precision_cap()
20
In :trac:`14084`, the category framework has been implemented for p-adic rings::
sage: TestSuite(R).run()
sage: K = Qp(7)
sage: TestSuite(K).run()
TESTS::
sage: R = Zp(5, 5, 'fixed-mod')
sage: R._repr_option('element_is_atomic')
False
"""
self._prec = prec
self.Element = element_class
default_category = getattr(self, '_default_category', None)
if self.is_field():
category = CompleteDiscreteValuationFields()
else:
category = CompleteDiscreteValuationRings()
if default_category is not None:
category = check_default_category(default_category, category)
Parent.__init__(self, base, names=(names,), normalize=False, category=category, element_constructor=element_class)
class PowerSeriesRing_over_field(PowerSeriesRing_domain):
_default_category = CompleteDiscreteValuationRings()
def fraction_field(self):
"""
Return the fraction field of this power series ring, which is
defined since this is over a field.
This fraction field is just the Laurent series ring over the base
field.
EXAMPLES::
sage: R.<t> = PowerSeriesRing(GF(7))
sage: R.fraction_field()
Laurent Series Ring in t over Finite Field of size 7
sage: Frac(R)
Laurent Series Ring in t over Finite Field of size 7
"""
return self.laurent_series_ring()
from sage.rings.integer import Integer
from sage.rings.finite_rings.finite_field_base import is_FiniteField
from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
from sage.misc.cachefunc import weak_cached_function
from sage.categories.fields import Fields
_Fields = Fields()
from sage.categories.commutative_rings import CommutativeRings
_CommutativeRings = CommutativeRings()
from sage.categories.complete_discrete_valuation import CompleteDiscreteValuationRings, CompleteDiscreteValuationFields
_CompleteDiscreteValuationRings = CompleteDiscreteValuationRings()
_CompleteDiscreteValuationFields = CompleteDiscreteValuationFields()
import sage.misc.weak_dict
_cache = sage.misc.weak_dict.WeakValueDictionary()
# The signature for this function is too complicated to express sensibly
# in any other way besides *args and **kwds (in Python 3 or Cython, we
# could probably do better thanks to PEP 3102).
def PolynomialRing(base_ring, *args, **kwds):
r"""
Return the globally unique univariate or multivariate polynomial
ring with given properties and variable name or names.