Originally Posted by

**AlexP** Thank you for the answer. I'd like to make sure I understand this. So $\displaystyle \langle f(x) \rangle = \{g(x)f(x)\,|\,g(x) \in F[x]\}$. Modding by this ideal yields all polynomials with degree less than that of $\displaystyle f(x)$. 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 $\displaystyle f(x)$ as a factor? I'm just missing something (obviously).