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 LaurentSeriesRing_field(LaurentSeriesRing_generic, ring.Field):
"""
Laurent series ring over a field.
TESTS::
sage: TestSuite(LaurentSeriesRing(QQ,'t')).run()
"""
_default_category = CompleteDiscreteValuationFields()
def __init__(self, power_series):
"""
Initialization
EXAMPLES::
sage: K.<q> = LaurentSeriesRing(QQ, default_prec=4); K
Laurent Series Ring in q over Rational Field
sage: 1 / (q-q^2)
q^-1 + 1 + q + q^2 + O(q^3)
sage: RZZ = LaurentSeriesRing(ZZ, 't')
sage: RZZ.category()
Category of infinite commutative no zero divisors algebras over (euclidean domains and infinite enumerated sets and metric spaces)
sage: TestSuite(RZZ).run()
sage: R1 = LaurentSeriesRing(Zmod(1), 't')
sage: R1.category()
Category of finite commutative algebras over (finite commutative rings and subquotients of monoids and quotients of semigroups and finite enumerated sets)
sage: TestSuite(R1).run()
sage: R2 = LaurentSeriesRing(Zmod(2), 't')
sage: R2.category()
Join of Category of complete discrete valuation fields and Category of commutative algebras over (finite enumerated fields and subquotients of monoids and quotients of semigroups) and Category of infinite sets
sage: TestSuite(R2).run()
sage: R4 = LaurentSeriesRing(Zmod(4), 't')
sage: R4.category()
Category of infinite commutative algebras over (finite commutative rings and subquotients of monoids and quotients of semigroups and finite enumerated sets)
sage: TestSuite(R4).run()
sage: RQQ = LaurentSeriesRing(QQ, 't')
sage: RQQ.category()
Join of Category of complete discrete valuation fields and Category of commutative algebras over (number fields and quotient fields and metric spaces) and Category of infinite sets
sage: TestSuite(RQQ).run()
"""
base_ring = power_series.base_ring()
category = Algebras(base_ring.category())
if base_ring in Fields():
category &= CompleteDiscreteValuationFields()
elif base_ring in IntegralDomains():
category &= IntegralDomains()
elif base_ring in Rings().Commutative():
category = category.Commutative()
if base_ring.is_zero():
category = category.Finite()
else:
category = category.Infinite()
self._power_series_ring = power_series
self._one_element = self.element_class(self, power_series.one())
CommutativeRing.__init__(self,
base_ring,
names=power_series.variable_names(),
category=category)
class LaurentSeriesRing_field(LaurentSeriesRing_generic, ring.Field):
_default_category = CompleteDiscreteValuationFields()
def __init__(self, base_ring, name=None, default_prec=None, sparse=False):
"""
Initialization
TESTS::
sage: TestSuite(LaurentSeriesRing(QQ,'t')).run()
"""
LaurentSeriesRing_generic.__init__(self, base_ring, name, default_prec, sparse)
def __init__(self, power_series):
"""
Initialization
EXAMPLES::
sage: K.<q> = LaurentSeriesRing(QQ, default_prec=4); K
Laurent Series Ring in q over Rational Field
sage: 1 / (q-q^2)
q^-1 + 1 + q + q^2 + O(q^3)
sage: RZZ = LaurentSeriesRing(ZZ, 't')
sage: RZZ.category()
Category of infinite integral domains
sage: TestSuite(RZZ).run()
sage: R1 = LaurentSeriesRing(Zmod(1), 't')
sage: R1.category()
Category of finite commutative rings
sage: TestSuite(R1).run()
sage: R2 = LaurentSeriesRing(Zmod(2), 't')
sage: R2.category()
Category of infinite complete discrete valuation fields
sage: TestSuite(R2).run()
sage: R4 = LaurentSeriesRing(Zmod(4), 't')
sage: R4.category()
Category of infinite commutative rings
sage: TestSuite(R4).run()
sage: RQQ = LaurentSeriesRing(QQ, 't')
sage: RQQ.category()
Category of infinite complete discrete valuation fields
sage: TestSuite(RQQ).run()
"""
base_ring = power_series.base_ring()
if base_ring in Fields():
category = CompleteDiscreteValuationFields()
elif base_ring in IntegralDomains():
category = IntegralDomains()
elif base_ring in Rings().Commutative():
category = Rings().Commutative()
else:
raise ValueError('unrecognized base ring')
if base_ring.is_zero():
category = category.Finite()
else:
category = category.Infinite()
CommutativeRing.__init__(self,
base_ring,
names=power_series.variable_names(),
category=category)
self._power_series_ring = power_series
def __init__(self, base_ring, names, sparse=True, category=None):
"""
Initialize ``self``.
TESTS::
sage: L = LazyLaurentSeriesRing(ZZ, 't')
sage: elts = L.some_elements()[:-2] # skip the non-exact elements
sage: TestSuite(L).run(elements=elts, skip=['_test_elements', '_test_associativity', '_test_distributivity', '_test_zero'])
sage: L.category()
Category of infinite commutative no zero divisors algebras over
(euclidean domains and infinite enumerated sets and metric spaces)
sage: L = LazyLaurentSeriesRing(QQ, 't')
sage: L.category()
Join of Category of complete discrete valuation fields
and Category of commutative algebras over (number fields and quotient fields and metric spaces)
and Category of infinite sets
sage: L = LazyLaurentSeriesRing(ZZ['x,y'], 't')
sage: L.category()
Category of infinite commutative no zero divisors algebras over
(unique factorization domains and commutative algebras over
(euclidean domains and infinite enumerated sets and metric spaces)
and infinite sets)
sage: E.<x,y> = ExteriorAlgebra(QQ)
sage: L = LazyLaurentSeriesRing(E, 't') # not tested
"""
self._sparse = sparse
self._coeff_ring = base_ring
# We always use the dense because our CS_exact is implemented densely
self._laurent_poly_ring = LaurentPolynomialRing(base_ring, names)
self._internal_poly_ring = self._laurent_poly_ring
category = Algebras(base_ring.category())
if base_ring in Fields():
category &= CompleteDiscreteValuationFields()
else:
if "Commutative" in base_ring.category().axioms():
category = category.Commutative()
if base_ring in IntegralDomains():
category &= IntegralDomains()
if base_ring.is_zero():
category = category.Finite()
else:
category = category.Infinite()
Parent.__init__(self, base=base_ring, names=names, category=category)
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.
There are many ways to specify the variables for the polynomial ring:
class PuiseuxSeriesRing_field(PuiseuxSeriesRing_generic, Field):
_default_category = CompleteDiscreteValuationFields()
def __init__(self, base_ring, name=None, default_prec=None, sparse=False):
PuiseuxSeriesRing_generic.__init__(self, base_ring, name, default_prec, sparse)