Commutative rings¶
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class
sage.categories.commutative_rings.
CommutativeRings
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton
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
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class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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extra_super_categories
()¶ 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
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class
CommutativeRings.
ElementMethods
¶
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class
CommutativeRings.
Finite
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton
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
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class
ParentMethods
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cyclotomic_cosets
(q, cosets=None)¶ 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
\[M_s(x) = \prod_{i \in C_s} (x - z^i),\]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 callled 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 ringcosets
– an optional lists of elements ofself
. If provided, the function only return the list of cosets that contain some element fromcosets
.
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 article Difference_set and
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), (3, 5), (2, 11), (6, 13), (4, 9), (5, 3)]]
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