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
ElementMethods
¶
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class
CommutativeRings.
Finite
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton
TESTS:
sage: C = Sets.Finite(); C Category of finite sets sage: type(C) <class 'sage.categories.finite_sets.FiniteSets_with_category'> sage: type(C).__base__.__base__ <class 'sage.categories.category_with_axiom.CategoryWithAxiom_singleton'> sage: TestSuite(C).run()
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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
be a finite commutative ring. The group of invertible elements
in
gives rise to a group action on
by multiplication. An orbit of the subgroup generated by an invertible element
is called a
-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
of
. The minimal polynomial of
over
is given by
where
is the
-cyclotomic coset mod
containing
,
.
Note
When
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). 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]
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