Ideals in Univariate Polynomial Rings

AUTHORS:

• David Roe (2009-12-14) – initial version.

class sage.rings.polynomial.ideal.Ideal_1poly_field(ring, gen)

Bases: Ideal_pid

An ideal in a univariate polynomial ring over a field.

groebner_basis(algorithm=None)

Return a Gröbner basis for this ideal.

The Gröbner basis has 1 element, namely the generator of the ideal. This trivial method exists for compatibility with multi-variate polynomial rings.

INPUT:

• algorithm – ignored

EXAMPLES:

sage: R.<x> = QQ[]
sage: I = R.ideal([x^2 - 1, x^3 - 1])
sage: G = I.groebner_basis(); G
[x - 1]
sage: type(G)
<class 'sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic'>
sage: list(G)
[x - 1]

residue_class_degree()

Returns the degree of the generator of this ideal.

This function is included for compatibility with ideals in rings of integers of number fields.

EXAMPLES:

sage: R.<t> = GF(5)[]
sage: P = R.ideal(t^4 + t + 1)
sage: P.residue_class_degree()
4

residue_field(names=None, check=True)

If this ideal is \(P \subset F_p[t]\), returns the quotient \(F_p[t]/P\).

EXAMPLES:

sage: R.<t> = GF(17)[]; P = R.ideal(t^3 + 2*t + 9)
sage: k.<a> = P.residue_field(); k
Residue field in a of Principal ideal (t^3 + 2*t + 9) of Univariate Polynomial Ring in t over Finite Field of size 17