Commutative rings

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
class ElementMethods
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()
class ParentMethods
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 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 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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