Compute the nilradical of F[x]/(p(x))

Let F be a field and p(x) \in F[x]. Compute the nilradical of F[x]/(p(x)).

We begin with a lemma. (I think I already proved this, but I can’t find it.)

Lemma: Let R be a unique factorization domain and x = \prod p_i^{k_i} an element of R, where the p_i are distinct irreducibles. Then \mathcal{N}(R/(x)) = (\prod p_i)/(x). Proof: Suppose y + (x) \in \mathcal{N}(R/(x)). Then y^n \in (x) for some n. Certainly then each p_i must divide y, as otherwise no power of y is divisible by p_i. Hence y \in (\prod p_i). Conversely, we have (\prod p_i)^{\max k_i} \in (x), so that (\prod p_i) + (x) \in \mathcal{N}(R). \square

Now F[x] is a unique factorization domain, and the lemma applies to F[x]/(p(x)).

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