def polynomial_default_category(base_ring, multivariate):
"""
Choose an appropriate category for a polynomial ring.
INPUT:
- ``base_ring``: The ring over which the polynomial ring shall be defined.
- ``multivariate``: Will the polynomial ring be multivariate?
EXAMPLES::
sage: QQ['t'].category() is Category.join([EuclideanDomains(), CommutativeAlgebras(QQ)])
True
sage: QQ['s','t'].category() is Category.join([UniqueFactorizationDomains(), CommutativeAlgebras(QQ)])
True
sage: QQ['s']['t'].category() is Category.join([UniqueFactorizationDomains(), CommutativeAlgebras(QQ['s'])])
True
"""
if base_ring in _Fields:
if multivariate:
return JoinCategory(
(_UniqueFactorizationDomains, CommutativeAlgebras(base_ring)))
return JoinCategory(
(_EuclideanDomains, CommutativeAlgebras(base_ring)))
if base_ring in _UFD: #base_ring.is_unique_factorization_domain():
return JoinCategory(
(_UniqueFactorizationDomains, CommutativeAlgebras(base_ring)))
if base_ring in _ID: #base_ring.is_integral_domain():
return JoinCategory((_IntegralDomains, CommutativeAlgebras(base_ring)))
if base_ring in _CommutativeRings: #base_ring.is_commutative():
return CommutativeAlgebras(base_ring)
return Algebras(base_ring)
sage: is_IntegerModRing(GF(13))
True
sage: is_IntegerModRing(GF(4, 'a'))
False
sage: is_IntegerModRing(10)
False
sage: is_IntegerModRing(ZZ)
False
"""
return isinstance(x, IntegerModRing_generic)
from sage.categories.commutative_rings import CommutativeRings
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.categories.category import JoinCategory
default_category = JoinCategory((CommutativeRings(), FiniteEnumeratedSets()))
ZZ = integer_ring.IntegerRing()
def _unit_gens_primepowercase(p, r):
r"""
Return a list of generators for `(\ZZ/p^r\ZZ)^*` and their orders.
EXAMPLES::
sage: from sage.rings.finite_rings.integer_mod_ring import _unit_gens_primepowercase
sage: _unit_gens_primepowercase(2, 3)
[(7, 2), (5, 2)]
sage: _unit_gens_primepowercase(17, 1)
[(3, 16)]
sage: _unit_gens_primepowercase(3, 3)