What can one expect for the quotient ring $F[x]/(f(x))$ when $f$ is not irreducible over $F$?

Question

Suppose $F$ is a field. When $f(x)\in F[x]$ is irreducible over $F$, $F[x]/(f(x))$ is a field.

What can one say (theorem/propositions?) in general about "structures" of the quotient ring when $f$ is not irreducible?

Answer 1

Write $f=\Pi f_i^{n_i}$ where $f_i$ is irreducible, since $F[X]$ is anEuclidean domain, you can apply the Chinese remainder theorem and obtain:

$F[X]/(f)=\Pi_i F[X]/(f_i^{n_i})$.

http://www.math.ru.nl/~bosma/onderwijs/voorjaar10/ca2010p4.pdf