Commutative rings#
- class sage.categories.commutative_rings.CommutativeRings(base_category)#
Bases:
CategoryWithAxiom_singletonThe 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), # optional - sage.rings.finite_rings ....: QQ, GF(5)]) in CommutativeRings() True
- class ElementMethods#
Bases:
object
- class Finite(base_category)#
Bases:
CategoryWithAxiom_singletonCheck that Sage knows that Cartesian products of finite commutative rings is a finite commutative ring.
EXAMPLES:
sage: cartesian_product([Zmod(34), # optional - sage.rings.finite_rings ....: GF(5)]) in Rings().Commutative().Finite() True
- class ParentMethods#
Bases:
object- cyclotomic_cosets(q, cosets=None)#
Return the (multiplicative) orbits of
qin 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 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') # optional - sage.rings.finite_rings sage: a = K.multiplicative_generator() # optional - sage.rings.finite_rings sage: K.cyclotomic_cosets(a**2, cosets=[1]) # optional - sage.rings.finite_rings [[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()) # optional - sage.rings.finite_rings [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') # optional - sage.rings.finite_rings sage: a = K.multiplicative_generator() # optional - sage.rings.finite_rings sage: R.<X> = PolynomialRing(K, 'X') # optional - sage.rings.finite_rings sage: a.minimal_polynomial('X') # optional - sage.rings.finite_rings X^3 + 2*X + 1 sage: cyc3 = Zmod(26).cyclotomic_cosets(3, cosets=[1]); cyc3 # optional - sage.rings.finite_rings [[1, 3, 9]] sage: prod(X - a**i for i in cyc3[0]) # optional - sage.rings.finite_rings X^3 + 2*X + 1 sage: (a**7).minimal_polynomial('X') # optional - sage.rings.finite_rings X^3 + X^2 + 2*X + 1 sage: cyc7 = Zmod(26).cyclotomic_cosets(3, cosets=[7]); cyc7 # optional - sage.rings.finite_rings [[7, 11, 21]] sage: prod(X - a**i for i in cyc7[0]) # optional - 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) # optional - sage.rings.finite_rings sage: a = K.multiplicative_generator() # optional - sage.rings.finite_rings sage: H = K.cyclotomic_cosets(a**2, cosets=[1, 2]); H # optional - 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) # optional - sage.rings.finite_rings [1, 2, 3, 4] sage: K = GF(37) # optional - sage.rings.finite_rings sage: a = K.multiplicative_generator() # optional - sage.rings.finite_rings sage: H = K.cyclotomic_cosets(a**4, cosets=[1]); H # optional - 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) # optional - sage.rings.finite_rings [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ..., 33, 34, 34, 35, 35, 36, 36]
The method
cyclotomic_cosetsworks on any finite commutative ring:sage: R = cartesian_product([GF(7), Zmod(14)]) # optional - sage.rings.finite_rings sage: a = R((3,5)) # optional - sage.rings.finite_rings sage: R.cyclotomic_cosets((3,5), [(1,1)]) # optional - sage.rings.finite_rings [[(1, 1), (2, 11), (3, 5), (4, 9), (5, 3), (6, 13)]]
- class ParentMethods#
Bases:
object- 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 orNone(default:None); the base of this extension or its defining morphismgen– a generator of this extension (over its base) orNone(default:None);gens– a list of generators of this extension (over its base) orNone(default:None);name– a variable name orNone(default:None)names– a list or a tuple of variable names orNone(default:None)
EXAMPLES:
We construct an extension of finite fields:
sage: F = GF(5^2) # optional - sage.rings.finite_rings sage: k = GF(5^4) # optional - sage.rings.finite_rings sage: z4 = k.gen() # optional - sage.rings.finite_rings sage: K = k.over(F) # optional - sage.rings.finite_rings sage: K # optional - sage.rings.finite_rings 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() # optional - sage.rings.finite_rings z4 sage: K(z4) # optional - sage.rings.finite_rings z4
However, it is possible to specify another generator and/or another name. For example:
sage: Ka = k.over(F, name='a') # optional - sage.rings.finite_rings sage: Ka # optional - sage.rings.finite_rings Field in a with defining polynomial x^2 + (4*z2 + 3)*x + z2 over its base sage: Ka.gen() # optional - sage.rings.finite_rings a sage: Ka(z4) # optional - sage.rings.finite_rings a sage: Kb = k.over(F, gen=-z4+1, name='b') # optional - sage.rings.finite_rings sage: Kb # optional - sage.rings.finite_rings Field in b with defining polynomial x^2 + z2*x + 4 over its base sage: Kb.gen() # optional - sage.rings.finite_rings b sage: Kb(-z4+1) # optional - sage.rings.finite_rings b
Note that the shortcut
K.<a>is also available:sage: KKa.<a> = k.over(F) # optional - sage.rings.finite_rings sage: KKa is Ka # optional - sage.rings.finite_rings True
Building an extension on top of another extension is allowed:
sage: L = GF(5^12).over(K) # optional - sage.rings.finite_rings sage: L # optional - 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() # optional - 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() # optional - 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
baseis omitted, the canonical base of the ring is used:sage: S.<x> = QQ[] sage: E = S.over() # optional - sage.modules sage: E # optional - sage.modules Univariate Polynomial Ring in x over Rational Field over its base sage: E.base_ring() # optional - sage.modules Rational Field
Here is an example where
baseis a defining morphism:sage: k.<a> = QQ.extension(x^2 - 2) # optional - sage.rings.number_field sage: l.<b> = QQ.extension(x^4 - 2) # optional - sage.rings.number_field sage: f = k.hom([b^2]) # optional - sage.rings.number_field sage: L = l.over(f) # optional - sage.rings.number_field sage: L # optional - sage.rings.number_field Field in b with defining polynomial x^2 - a over its base sage: L.base_ring() # optional - sage.rings.number_field Number Field in a with defining polynomial x^2 - 2
Similarly, one can create a tower of extensions:
sage: K = k.over() # optional - sage.rings.number_field sage: L = l.over(Hom(K, l)(f)) # optional - sage.rings.number_field sage: L # optional - sage.rings.number_field Field in b with defining polynomial x^2 - a over its base sage: L.base_ring() # optional - sage.rings.number_field Field in a with defining polynomial x^2 - 2 over its base sage: L.bases() # optional - sage.rings.number_field [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]