Congruences classes quotient ring

asked 2021-11-20 22:58:40 +0100

schwabis gravatar image

I am playing around with quotient rings in several variables and would like to get the congruence classes. How would I do that?

Let's say I have:

R.<x,y> = ZZ[]
S.<a,b> = R.quotient((x^2 + y^2, 17))

How do I get all congruence classes of R/[x^2 + y^2, 17]?

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There are infinitely many equivalence classes here - eg. all $x^k$ belong to distinct classes. So please clarify what you mean under "get all congruence classes"?

Max Alekseyev gravatar imageMax Alekseyev ( 2021-11-21 04:24:29 +0100 )edit

If you have a homogeneous ideal in a polynomial ring over a field then you can do e.g. R.<x,y> = GF(17)[]; I = R.ideal((x^2 + y^2)); I.normal_basis(4) to get basis elements of degree 4.

rburing gravatar imagerburing ( 2021-11-21 10:07:17 +0100 )edit