Lemma 57.11.2. Let $X$ be a normal integral scheme with function field $K$. Let $Y \to X$ be a finite étale morphism. If $Y$ is connected, then $Y$ is an integral normal scheme and $Y$ is the normalization of $X$ in the function field of $Y$.
Proof. The scheme $Y$ is normal by Descent, Lemma 35.15.2. Since $Y \to X$ is flat every generic point of $Y$ maps to the generic point of $X$ by Morphisms, Lemma 29.25.9. Since $Y \to X$ is finite we see that $Y$ has a finite number of irreducible components. Thus $Y$ is the disjoint union of a finite number of integral normal schemes by Properties, Lemma 28.7.5. Thus if $Y$ is connected, then $Y$ is an integral normal scheme.
Let $L$ be the function field of $Y$ and let $Y' \to X$ be the normalization of $X$ in $L$. By Morphisms, Lemma 29.53.4 we obtain a factorization $Y' \to Y \to X$ and $Y' \to Y$ is the normalization of $Y$ in $L$. Since $Y$ is normal it is clear that $Y' = Y$ (this can also be deduced from Morphisms, Lemma 29.54.8). $\square$
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