Ideals in Univariate Polynomial Rings#
AUTHORS:
David Roe (2009-12-14) – initial version.
- class sage.rings.polynomial.ideal.Ideal_1poly_field(ring, gens, coerce=True, **kwds)#
Bases:
Ideal_pid
An ideal in a univariate polynomial ring over a field.
- change_ring(R)#
Coerce an ideal into a new ring.
EXAMPLES:
sage: R.<q> = QQ[] sage: I = R.ideal([q^2+q-1]) sage: I.change_ring(RR['q']) Principal ideal (q^2 + q - 1.00000000000000) of Univariate Polynomial Ring in q over Real Field with 53 bits of precision
- 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()#
Return 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 \(P \subset F_p[t]\), return 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 # needs sage.rings.finite_rings Residue field in a of Principal ideal (t^3 + 2*t + 9) of Univariate Polynomial Ring in t over Finite Field of size 17