The Stacks project

Lemma 15.104.15. Let $L/K$ be an extension of fields. If $L \otimes _ K L \to L$ is flat, then $L$ is an algebraic separable extension of $K$.

Proof. By Lemma 15.104.10 we see that any subfield $K \subset L' \subset L$ the map $L' \otimes _ K L' \to L'$ is flat. Thus we may assume $L$ is a finitely generated field extension of $K$. In this case the fact that $L/K$ is formally unramified (Lemma 15.104.12) implies that $L/K$ is finite separable, see Algebra, Lemma 10.158.1. $\square$

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