Commutative rings#

class sage.categories.commutative_rings.CommutativeRings(base_category)#

Bases: 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]

class CartesianProducts(category, *args)#

Bases: CartesianProductsCategory

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

class ElementMethods#: Bases: object

class Finite(base_category)#

Bases: 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

class ParentMethods#

Bases: object

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 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: # needs sage.rings.finite_rings
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: # needs sage.rings.finite_rings
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])                               # needs sage.rings.finite_rings
X^3 + 2*X + 1
sage: (a**7).minimal_polynomial('X')                                # needs sage.rings.finite_rings
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])                               # needs sage.rings.finite_rings
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()                              # needs sage.libs.pari
sage: H = K.cyclotomic_cosets(a**2, cosets=[1, 2]); H               # needs sage.rings.finite_rings
[[1, 4], [2, 3]]
sage: sorted(x - y for D in H for x in D for y in D if x != y)      # needs sage.rings.finite_rings
[1, 2, 3, 4]

sage: K = GF(37)
sage: a = K.multiplicative_generator()                              # needs sage.libs.pari
sage: H = K.cyclotomic_cosets(a**4, cosets=[1]); H                  # needs sage.rings.finite_rings
[[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)      # needs sage.rings.finite_rings
[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)]]

class ParentMethods#

Bases: object

is_commutative()#

Return whether the ring is commutative.

The answer is True only if the category is a sub-category of CommutativeRings.

It is recommended to use instead R in Rings().Commutative().

EXAMPLES:

sage: QQ.is_commutative()
True
sage: QQ['x,y,z'].is_commutative()
True

over(base=None, gen=None, gens=None, name=None, names=None)#

Return this ring, considered as an extension of base.

INPUT:

base – a commutative ring or a morphism or None (default: None); the base of this extension or its defining morphism
gen – a generator of this extension (over its base) or None (default: None);
gens – a list of generators of this extension (over its base) or None (default: None);
name – a variable name or None (default: None)
names – a list or a tuple of variable names or None (default: None)

EXAMPLES:

We construct an extension of finite fields:

sage: # needs sage.rings.finite_rings
sage: F = GF(5^2)
sage: k = GF(5^4)
sage: z4 = k.gen()
sage: K = k.over(F); K                                                  # needs sage.modules
Field in z4 with defining polynomial
 x^2 + (4*z2 + 3)*x + z2 over its base

If not explicitly given, the default generator of the top ring (here k) is used and the same name is kept:

sage: K.gen()                                                           # needs sage.modules sage.rings.finite_rings
z4
sage: K(z4)                                                             # needs sage.modules sage.rings.finite_rings
z4

However, it is possible to specify another generator and/or another name. For example:

sage: # needs sage.modules sage.rings.finite_rings
sage: Ka = k.over(F, name='a'); Ka
Field in a with defining polynomial
 x^2 + (4*z2 + 3)*x + z2 over its base
sage: Ka.gen()
a
sage: Ka(z4)
a

sage: # needs sage.modules sage.rings.finite_rings
sage: Kb = k.over(F, gen=-z4+1, name='b')
sage: Kb
Field in b with defining polynomial x^2 + z2*x + 4 over its base
sage: Kb.gen()
b
sage: Kb(-z4+1)
b

Note that the shortcut K.<a> is also available:

sage: KKa.<a> = k.over(F)                                               # needs sage.modules sage.rings.finite_rings
sage: KKa is Ka                                                         # needs sage.modules sage.rings.finite_rings
True

Building an extension on top of another extension is allowed:

sage: L = GF(5^12).over(K); L                                           # needs sage.modules sage.rings.finite_rings
Field in z12 with defining polynomial
 x^3 + (1 + (4*z2 + 2)*z4)*x^2 + (2 + 2*z4)*x - z4 over its base
sage: L.base_ring()                                                     # needs sage.modules sage.rings.finite_rings
Field in z4 with defining polynomial
 x^2 + (4*z2 + 3)*x + z2 over its base

The successive bases of an extension are accessible via the method sage.rings.ring_extension.RingExtension_generic.bases():

sage: L.bases()                                                         # needs sage.modules sage.rings.finite_rings
[Field in z12 with defining polynomial
  x^3 + (1 + (4*z2 + 2)*z4)*x^2 + (2 + 2*z4)*x - z4 over its base,
 Field in z4 with defining polynomial
  x^2 + (4*z2 + 3)*x + z2 over its base,
 Finite Field in z2 of size 5^2]

When base is omitted, the canonical base of the ring is used:

sage: S.<x> = QQ[]
sage: E = S.over(); E                                                   # needs sage.modules
Univariate Polynomial Ring in x over Rational Field over its base
sage: E.base_ring()                                                     # needs sage.modules
Rational Field

Here is an example where base is a defining morphism:

sage: # needs sage.modules sage.rings.number_field
sage: k.<a> = QQ.extension(x^2 - 2)
sage: l.<b> = QQ.extension(x^4 - 2)
sage: f = k.hom([b^2])
sage: L = l.over(f)
sage: L
Field in b with defining polynomial x^2 - a over its base
sage: L.base_ring()
Number Field in a with defining polynomial x^2 - 2

Similarly, one can create a tower of extensions:

sage: # needs sage.modules sage.rings.number_field
sage: K = k.over()
sage: L = l.over(Hom(K, l)(f)); L
Field in b with defining polynomial x^2 - a over its base
sage: L.base_ring()
Field in a with defining polynomial x^2 - 2 over its base
sage: L.bases()
[Field in b with defining polynomial x^2 - a over its base,
 Field in a with defining polynomial x^2 - 2 over its base,
 Rational Field]