Coverage for local/lib/python2.7/site-packages/sage/rings/polynomial/ideal.py : 94%

# -*- coding: utf-8 -*- 

""" 

Ideals in Univariate Polynomial Rings. 

AUTHORS: 

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

""" 

#***************************************************************************** 

# Copyright (C) 2009 David Roe <roed@math.harvard.edu> 

# William Stein <wstein@gmail.com> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.rings.ideal import Ideal_pid 

class Ideal_1poly_field(Ideal_pid): 

""" 

An ideal in a univariate polynomial ring over a field. 

""" 

def residue_class_degree(self): 

""" 

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 

""" 

return self.gen().degree() 

def residue_field(self, 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 

""" 

if check: 

if not self.ring().base_ring().is_finite(): 

raise TypeError("residue fields only supported for polynomial rings over finite fields.") 

if not self.is_prime(): 

raise ValueError("%s is not a prime ideal"%self) 

from sage.rings.finite_rings.residue_field import ResidueField 

return ResidueField(self, names, check=False) 

def groebner_basis(self, 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] 

""" 

gb = self.gens_reduced() 

from sage.rings.polynomial.multi_polynomial_sequence import PolynomialSequence_generic 

return PolynomialSequence_generic([gb], self.ring(), immutable=True)