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

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?

• If it isn't irreducible, you can spect to get zero divisors. It means that your new ring won't become a domain. – iam_agf Nov 20 '16 at 3:45
• You get a finite product of Artinian, local, self injective, uniserial rings (as long as $f$ isn't a constant.) – rschwieb Nov 20 '16 at 12:17

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:
$F[X]/(f)=\Pi_i F[X]/(f_i^{n_i})$.
• ...If the $f_i$ are pairwise non associated. – Mariano Suárez-Álvarez Nov 20 '16 at 3:58