Proposition 41.19.7. Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an étale homomorphism of local rings. Then $A$ is a normal domain if and only if $B$ is so.
[Expose I, Theorem 9.5 part (i), SGA1]
Proof.
See Algebra, Lemma 10.164.3 for descending normality. Conversely, assume $A$ is normal. By assumption $B$ is a localization of a finite type $A$-algebra $B'$ at some prime $\mathfrak q$. After replacing $B'$ by a localization we may assume that $B'$ is étale over $A$, see Lemma 41.11.2. Then we see that Algebra, Lemma 10.163.9 applies to $A \to B'$ and we conclude that $B'$ is normal. Hence $B$ is a normal domain.
$\square$
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