Lemma 15.40.6. Let $k \subset K$ be a field extension. Then $k \to K$ is a regular ring map if and only if $K$ is a separable field extension of $k$.

**Proof.**
If $k \to K$ is regular, then $K$ is geometrically reduced over $k$, hence $K$ is separable over $k$ by Algebra, Proposition 10.152.9. Conversely, if $K/k$ is separable, then $K$ is a colimit of smooth $k$-algebras, see Algebra, Lemma 10.152.11 hence is regular by Lemma 15.40.5.
$\square$

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