Lemma 15.44.4. If $A \to B$ is an étale ring map and $A$ is a Dedekind domain, then $B$ is a finite product of Dedekind domains. In particular, the localizations $B_\mathfrak q$ for $\mathfrak q \subset B$ maximal are discrete valuation rings.
Proof. The statement on the local rings follows from Lemmas 15.44.2 and 15.44.3 and Algebra, Lemma 10.119.7. It follows that $B$ is a Noetherian normal ring of dimension $1$. By Algebra, Lemma 10.37.16 we conclude that $B$ is a finite product of normal domains of dimension $1$. These are Dedekind domains by Algebra, Lemma 10.120.17. $\square$
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