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.

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