Thank you for the answer. I'd like to make sure I understand this. So

. Modding by this ideal yields all polynomials with degree less than that of

. But then why exactly doesn't it matter which polynomial (ideal) we mod by? Modding by a given polynomial makes all multiples of this polynomial "0," but then what about the other irreducible, say, quadratics? I know this isn't well-posed at all but hopefully someone will see what I'm trying to ask...

I guess I could also ask, what happens to elements in the other ideals generated by the other irreducible quadratics, that do not have

as a factor? I'm just missing something (obviously).