The Stacks project

Lemma 10.47.13. Let $K/k$ be a field extension. Consider the subextension $K/k'/k$ consisting of elements separably algebraic over $k$. Then $K$ is geometrically irreducible over $k'$. If $K/k$ is a finitely generated field extension, then $[k' : k] < \infty $.

Proof. The first statement is immediate from Lemma 10.47.12 and the fact that elements separably algebraic over $k'$ are in $k'$ by the transitivity of separable algebraic extensions, see Fields, Lemma 9.12.12. If $K/k$ is finitely generated, then $k'$ is finite over $k$ by Fields, Lemma 9.26.11. $\square$

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