def __init__(self, order, cache=None, category=None):
"""
Create with the command ``IntegerModRing(order)``.
TESTS::
sage: FF = IntegerModRing(29)
sage: TestSuite(FF).run()
sage: F19 = IntegerModRing(19, category = Fields())
sage: TestSuite(F19).run()
sage: F23 = IntegerModRing(23)
sage: F23 in Fields()
True
sage: TestSuite(F23).run()
sage: Z16 = IntegerModRing(16)
sage: TestSuite(Z16).run()
sage: R = Integers(100000)
sage: TestSuite(R).run() # long time (17s on sage.math, 2011)
"""
order = ZZ(order)
if order <= 0:
raise ZeroDivisionError("order must be positive")
self.__order = order
self._pyx_order = integer_mod.NativeIntStruct(order)
if category is None:
from sage.categories.commutative_rings import CommutativeRings
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.categories.category import Category
category = Category.join(
[CommutativeRings(),
FiniteEnumeratedSets()])
# category = default_category
# If the category is given then we trust that is it right.
# Give the generator a 'name' to make quotients work. The
# name 'x' is used because it's also used for the ring of
# integers: see the __init__ method for IntegerRing_class in
# sage/rings/integer_ring.pyx.
quotient_ring.QuotientRing_generic.__init__(self,
ZZ,
ZZ.ideal(order),
names=('x', ),
category=category)
# Calling ParentWithGens is not needed, the job is done in
# the quotient ring initialisation.
#ParentWithGens.__init__(self, self, category = category)
# We want that the ring is its own base ring.
self._base = self
if cache is None:
cache = order < 500
if cache:
self._precompute_table()
self._zero_element = integer_mod.IntegerMod(self, 0)
self._one_element = integer_mod.IntegerMod(self, 1)
def __init__(self, order, cache=None, category=None):
"""
Create with the command ``IntegerModRing(order)``.
TESTS::
sage: FF = IntegerModRing(29)
sage: TestSuite(FF).run()
sage: F19 = IntegerModRing(19, is_field=True)
sage: TestSuite(F19).run()
sage: F23 = IntegerModRing(23)
sage: F23 in Fields()
True
sage: TestSuite(F23).run()
sage: Z16 = IntegerModRing(16)
sage: TestSuite(Z16).run()
sage: R = Integers(100000)
sage: TestSuite(R).run() # long time (17s on sage.math, 2011)
"""
order = ZZ(order)
if order <= 0:
raise ZeroDivisionError("order must be positive")
self.__order = order
self._pyx_order = integer_mod.NativeIntStruct(order)
global default_category
if category is None:
category = default_category
else:
# If the category is given, e.g., as Fields(), then we still
# know that the result will also live in default_category.
# Hence, we use the join of the default and the given category.
category = category.join([category, default_category])
# Give the generator a 'name' to make quotients work. The
# name 'x' is used because it's also used for the ring of
# integers: see the __init__ method for IntegerRing_class in
# sage/rings/integer_ring.pyx.
quotient_ring.QuotientRing_generic.__init__(self,
ZZ,
ZZ.ideal(order),
names=('x', ),
category=category)
# We want that the ring is its own base ring.
self._base = self
if cache is None:
cache = order < 500
if cache:
self._precompute_table()
self._zero_element = integer_mod.IntegerMod(self, 0)
self._one_element = integer_mod.IntegerMod(self, 1)
def __init__(self, order, cache=None, category=None):
"""
Create with the command IntegerModRing(order)
INPUT:
- ``order`` - an integer 1
- ``category`` - a subcategory of ``CommutativeRings()`` (the default)
(currently only available for subclasses)
OUTPUT:
- ``IntegerModRing`` - the ring of integers modulo
N.
EXAMPLES:
First we compute with integers modulo `29`.
::
sage: FF = IntegerModRing(29)
sage: FF
Ring of integers modulo 29
sage: FF.category()
Join of Category of commutative rings and Category of subquotients of monoids and Category of quotients of semigroups and Category of finite enumerated sets
sage: FF.is_field()
True
sage: FF.characteristic()
29
sage: FF.order()
29
sage: gens = FF.unit_gens()
sage: a = gens[0]
sage: a
2
sage: a.is_square()
False
sage: def pow(i): return a**i
sage: [pow(i) for i in range(16)]
[1, 2, 4, 8, 16, 3, 6, 12, 24, 19, 9, 18, 7, 14, 28, 27]
sage: TestSuite(FF).run()
We have seen above that an integer mod ring is, by default, not
initialised as an object in the category of fields. However, one
can force it to be. Moreover, testing containment in the category
of fields my re-initialise the category of the integer mod ring::
sage: F19 = IntegerModRing(19, category = Fields())
sage: F19.category().is_subcategory(Fields())
True
sage: F23 = IntegerModRing(23)
sage: F23.category().is_subcategory(Fields())
False
sage: F23 in Fields()
True
sage: F23.category().is_subcategory(Fields())
True
sage: TestSuite(F19).run()
sage: TestSuite(F23).run()
Next we compute with the integers modulo `16`.
::
sage: Z16 = IntegerModRing(16)
sage: Z16.category()
Join of Category of commutative rings and Category of subquotients of monoids and Category of quotients of semigroups and Category of finite enumerated sets
sage: Z16.is_field()
False
sage: Z16.order()
16
sage: Z16.characteristic()
16
sage: gens = Z16.unit_gens()
sage: gens
[15, 5]
sage: a = gens[0]
sage: b = gens[1]
sage: def powa(i): return a**i
sage: def powb(i): return b**i
sage: gp_exp = FF.unit_group_exponent()
sage: gp_exp
28
sage: [powa(i) for i in range(15)]
[1, 15, 1, 15, 1, 15, 1, 15, 1, 15, 1, 15, 1, 15, 1]
sage: [powb(i) for i in range(15)]
[1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9]
sage: a.multiplicative_order()
2
sage: b.multiplicative_order()
4
sage: TestSuite(Z16).run()
Saving and loading::
sage: R = Integers(100000)
sage: TestSuite(R).run() # long time (17s on sage.math, 2011)
Testing ideals and quotients::
sage: Z10 = Integers(10)
sage: I = Z10.principal_ideal(0)
sage: Z10.quotient(I) == Z10
True
sage: I = Z10.principal_ideal(2)
sage: Z10.quotient(I) == Z10
False
sage: I.is_prime()
True
"""
order = ZZ(order)
if order <= 0:
raise ZeroDivisionError, "order must be positive"
self.__order = order
self._pyx_order = integer_mod.NativeIntStruct(order)
self.__unit_group_exponent = None
self.__factored_order = None
self.__factored_unit_order = None
if category is None:
from sage.categories.commutative_rings import CommutativeRings
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.categories.category import Category
category = Category.join(
[CommutativeRings(),
FiniteEnumeratedSets()])
# category = default_category
# If the category is given then we trust that is it right.
# Give the generator a 'name' to make quotients work. The
# name 'x' is used because it's also used for the ring of
# integers: see the __init__ method for IntegerRing_class in
# sage/rings/integer_ring.pyx.
quotient_ring.QuotientRing_generic.__init__(self,
ZZ,
ZZ.ideal(order),
names=('x', ),
category=category)
# Calling ParentWithGens is not needed, the job is done in
# the quotient ring initialisation.
#ParentWithGens.__init__(self, self, category = category)
# We want that the ring is its own base ring.
self._base = self
if cache is None:
cache = order < 500
if cache:
self._precompute_table()
self._zero_element = integer_mod.IntegerMod(self, 0)
self._one_element = integer_mod.IntegerMod(self, 1)