# How to get complete list of polynomials in an ideal, if its generating set is given?

Anonymous

I am using sage for working in polynomials in several variables. I want to ask, 1) if generating set of an ideal is given, then is there any command which will give me a complete list of polynomials in that ideal. 2) How to compute addition of two ideals in sage?

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(1) In general, an ideal is infinite, so it would not make sense to give a complete list.

Do you have a particular setting in mind where ideals are finite?

(2) To add ideals, just use the plus sign.

Example.

sage: R.<x, y, z> = QQ[]
sage: G = R.ideal(x^2)
sage: H = R.ideal([y^2, z*z])
sage: J = G + H
sage: J
Ideal (x^2, y^2, z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field

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I am thinking for an ideal in a polynomial ring in n variables over finite field with 2 elements

( 2016-12-03 14:19:18 +0200 )edit

@Nilesh -- still, an ideal would contain polynomials of arbitrary degree.

( 2016-12-03 14:56:43 +0200 )edit

Sir, if I work in a polynomial ring over a finite field with 2 elements in 35 variables, and find out reduced Grobner basis of that ideal, then will it be an ideal having finite generating set?(I think it has to be, by Hilbert basis theorem), I am unable to get that generating set.

( 2016-12-05 02:38:18 +0200 )edit