Lemma 10.152.10. Let $k \subset K$ be a finitely generated field extension. Then $K$ is separable over $k$ if and only if $K$ is the localization of a smooth $k$-algebra.
Choose a finite type $k$-algebra $R$ which is a domain whose fraction field is $K$. Lemma 10.138.9 says that $k \to R$ is smooth at $(0)$ if and only if $K/k$ is separable. This proves the lemma.
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