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commutative_rings.py
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commutative_rings.py
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r"""
Commutative rings
"""
#*****************************************************************************
# Copyright (C) 2005 David Kohel <kohel@maths.usyd.edu>
# William Stein <wstein@math.ucsd.edu>
# 2008 Teresa Gomez-Diaz (CNRS) <Teresa.Gomez-Diaz@univ-mlv.fr>
# 2008-2013 Nicolas M. Thiery <nthiery at users.sf.net>
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#******************************************************************************
from sage.categories.category_with_axiom import CategoryWithAxiom
from sage.categories.cartesian_product import CartesianProductsCategory
class CommutativeRings(CategoryWithAxiom):
"""
The category of commutative rings
commutative rings with unity, i.e. rings with commutative * and
a multiplicative identity
EXAMPLES::
sage: C = CommutativeRings(); C
Category of commutative rings
sage: C.super_categories()
[Category of rings, Category of commutative monoids]
TESTS::
sage: TestSuite(C).run()
sage: QQ['x,y,z'] in CommutativeRings()
True
sage: GroupAlgebra(DihedralGroup(3), QQ) in CommutativeRings()
False
sage: MatrixSpace(QQ,2,2) in CommutativeRings()
False
GroupAlgebra should be fixed::
sage: GroupAlgebra(CyclicPermutationGroup(3), QQ) in CommutativeRings() # todo: not implemented
True
"""
class ParentMethods:
def _test_divides(self, **options):
r"""
Run generic tests on the method :meth:`divides`.
EXAMPLES::
sage: ZZ._test_divides()
"""
tester = self._tester(**options)
# 1. is there a divides method ?
a = self.an_element()
try:
a.divides
except AttributeError:
return
# 2. divisibility of 0 and 1
z = self.zero()
o = self.one()
tester.assertTrue(z.divides(z))
tester.assertTrue(o.divides(o))
tester.assertTrue(o.divides(z))
tester.assertTrue(z.divides(o) is self.is_zero())
if not self.is_exact():
return
# 3. divisibility of some elements
S = tester.some_elements()
for a,b in tester.some_elements(repeat=2):
try:
test = a.divides(a*b)
except NotImplementedError:
pass
else:
tester.assertTrue(test)
class ElementMethods:
pass
class Finite(CategoryWithAxiom):
r"""
Check that Sage knows that Cartesian products of finite commutative
rings is a finite commutative ring.
EXAMPLES::
sage: cartesian_product([Zmod(34), GF(5)]) in Rings().Commutative().Finite()
True
"""
class ParentMethods:
def cyclotomic_cosets(self, q, cosets=None):
r"""
Return the (multiplicative) orbits of ``q`` in the ring.
Let `R` be a finite commutative ring. The group of invertible
elements `R^*` in `R` gives rise to a group action on `R` by
multiplication. An orbit of the subgroup generated by an
invertible element `q` is called a `q`-*cyclotomic coset* (since
in a finite ring, each invertible element is a root of unity).
These cosets arise in the theory of minimal polynomials of
finite fields, duadic codes and combinatorial designs. Fix a
primitive element `z` of `GF(q^k)`. The minimal polynomial of
`z^s` over `GF(q)` is given by
.. MATH::
M_s(x) = \prod_{i \in C_s} (x - z^i),
where `C_s` is the `q`-cyclotomic coset mod `n` containing `s`,
`n = q^k - 1`.
.. NOTE::
When `R = \ZZ / n \ZZ` the smallest element of each coset is
sometimes called a *coset leader*. This function returns
sorted lists so that the coset leader will always be the
first element of the coset.
INPUT:
- ``q`` -- an invertible element of the ring
- ``cosets`` -- an optional lists of elements of ``self``. If
provided, the function only return the list of cosets that
contain some element from ``cosets``.
OUTPUT:
A list of lists.
EXAMPLES::
sage: Zmod(11).cyclotomic_cosets(2)
[[0], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]]
sage: Zmod(15).cyclotomic_cosets(2)
[[0], [1, 2, 4, 8], [3, 6, 9, 12], [5, 10], [7, 11, 13, 14]]
Since the group of invertible elements of a finite field is
cyclic, the set of squares is a particular case of cyclotomic
coset::
sage: K = GF(25,'z')
sage: a = K.multiplicative_generator()
sage: K.cyclotomic_cosets(a**2,cosets=[1])
[[1, 2, 3, 4, z + 1, z + 3,
2*z + 1, 2*z + 2, 3*z + 3,
3*z + 4, 4*z + 2, 4*z + 4]]
sage: sorted(b for b in K if not b.is_zero() and b.is_square())
[1, 2, 3, 4, z + 1, z + 3,
2*z + 1, 2*z + 2, 3*z + 3,
3*z + 4, 4*z + 2, 4*z + 4]
We compute some examples of minimal polynomials::
sage: K = GF(27,'z')
sage: a = K.multiplicative_generator()
sage: R.<X> = PolynomialRing(K, 'X')
sage: a.minimal_polynomial('X')
X^3 + 2*X + 1
sage: cyc3 = Zmod(26).cyclotomic_cosets(3,cosets=[1]); cyc3
[[1, 3, 9]]
sage: prod(X - a**i for i in cyc3[0])
X^3 + 2*X + 1
sage: (a**7).minimal_polynomial('X')
X^3 + X^2 + 2*X + 1
sage: cyc7 = Zmod(26).cyclotomic_cosets(3,cosets=[7]); cyc7
[[7, 11, 21]]
sage: prod(X - a**i for i in cyc7[0])
X^3 + X^2 + 2*X + 1
Cyclotomic cosets of fields are useful in combinatorial design
theory to provide so called difference families (see
:wikipedia:`Difference_set` and
:mod:`~sage.combinat.designs.difference_family`). This is
illustrated on the following examples::
sage: K = GF(5)
sage: a = K.multiplicative_generator()
sage: H = K.cyclotomic_cosets(a**2, cosets=[1,2]); H
[[1, 4], [2, 3]]
sage: sorted(x-y for D in H for x in D for y in D if x != y)
[1, 2, 3, 4]
sage: K = GF(37)
sage: a = K.multiplicative_generator()
sage: H = K.cyclotomic_cosets(a**4, cosets=[1]); H
[[1, 7, 9, 10, 12, 16, 26, 33, 34]]
sage: sorted(x-y for D in H for x in D for y in D if x != y)
[1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ..., 33, 34, 34, 35, 35, 36, 36]
The method ``cyclotomic_cosets`` works on any finite commutative
ring::
sage: R = cartesian_product([GF(7), Zmod(14)])
sage: a = R((3,5))
sage: R.cyclotomic_cosets((3,5), [(1,1)])
[[(1, 1), (2, 11), (3, 5), (4, 9), (5, 3), (6, 13)]]
"""
q = self(q)
try:
~q
except ZeroDivisionError:
raise ValueError("%s is not invertible in %s"%(q,self))
if cosets is None:
rest = set(self)
else:
rest = set(self(x) for x in cosets)
orbits = []
while rest:
x0 = rest.pop()
o = [x0]
x = q*x0
while x != x0:
o.append(x)
rest.discard(x)
x *= q
o.sort()
orbits.append(o)
orbits.sort()
return orbits
class CartesianProducts(CartesianProductsCategory):
def extra_super_categories(self):
r"""
Let Sage knows that Cartesian products of commutative rings is a
commutative ring.
EXAMPLES::
sage: CommutativeRings().Commutative().CartesianProducts().extra_super_categories()
[Category of commutative rings]
sage: cartesian_product([ZZ, Zmod(34), QQ, GF(5)]) in CommutativeRings()
True
"""
return [CommutativeRings()]