A question about fields and separability in Serre's "Local Fields"

Question

On page 14 of the English edition of Serre's "Local Fields", that is chapter 1, section 4, I am confused by the following; there is talk of fields $B/\mathfrak P$ and $A/\mathfrak p$ for prime ideals $\mathfrak P$ of a ring $B$ and $\mathfrak p$ of a ring $A.$ Surely that would require $\mathfrak P$ and $\mathfrak p$ to be maximal ideals? Also in the next paragraph there is talk of the extension $B/\mathfrak P$ of $A/\mathfrak p$ being separable, which is a concept I know only for field extensions. So how should I interpret all of this? Thanks for any help!

Answer 1

Right before that he says

Let us keep the hypotheses of prop. 9. If $\mathfrak{P}$ is a non-zero prime ideal of $B$...

Proposition 9 says

Proposition 9. If $A$ is Dedekind then $B$ is Dedekind.

and in a Dedekind domain, every non-zero prime ideal is maximal. Thus $\mathfrak{P}$ and $\mathfrak{p}=\mathfrak{P}\cap A$ are indeed maximal, and thus $B/\mathfrak{P}$ and $A/\mathfrak{p}$ are indeed fields.

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But I'll also note that there is a notion of separability for ring extensions; see The Separable Galois Theory of Commutative Rings for example.