Ideals in Univariate Polynomial Rings.¶
AUTHORS:
 David Roe (20091214) – initial version.

class
sage.rings.polynomial.ideal.
Ideal_1poly_field
(ring, gen)¶ Bases:
sage.rings.ideal.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 multivariate 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
