## Characterize the ideals of F[x]/(p(x))

Let $F$ be a field and let $p(x) \in F[x]$. Describe the ideals of $F[x]/(p(x))$ in terms of the factorization of $p(x)$.

We begin with a lemma.

Lemma: Let $R$ be a principal ideal domain and let $a \in R$ factor into irreducibles as $a = \prod p_i^{k_i}$. Then the ideals of $R/(a)$ have the form $(b)/(a)$ where $b = \prod p_i^{\ell_i}$ and $0 \leq \ell_i \leq k_i$. Proof: $R$ is a unique factorization domain. By the lattice isomorphism theorem for rings, the ideals of $R/(a)$ correspond to the ideals of $R$ which contain $a$, and these in turn correspond to the divisors of $a$. In this previous exercise, we characterized the divisors of $a$ in terms of its factorization as desired. $\square$

For this problem, since $F$ is a field, $F[x]$ is a Euclidean domain and thus a principal ideal domain.