Lemma 10.46.9. Let $k \subset K$ be a field extension. Consider the subextension $k \subset k' \subset K$ such that $k \subset k'$ is separable algebraic and $k' \subset K$ maximal with this property. Then $K$ is geometrically irreducible over $k'$. If $K/k$ is a finitely generated field extension, then $[k' : k] < \infty $.
Proof. Let $k'' \subset K$ be the algebraic closure of $k$ in $K$. By Lemma 10.46.8 we see that $K$ is geometrically irreducible over $k''$. Since $k' \subset k''$ is purely inseparable (Fields, Lemma 9.14.6) we see from Lemma 10.45.7 that the extension $k' \subset K$ is also geometrically irreducible. If $k \subset K$ is finitely generated, then $k'$ is finite over $k$ by Fields, Lemma 9.26.10. $\square$
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