The Stacks project

Lemma 15.14.7. Let $A$ be absolutely integrally closed. Let $\mathfrak p \subset A$ be a prime. Then the local ring $A_\mathfrak p$ is strictly henselian.

Proof. By Lemma 15.14.3 we may assume $A$ is a local ring and $\mathfrak p$ is its maximal ideal. The residue field is algebraically closed by Lemma 15.14.3. Every monic polynomial decomposes completely into linear factors hence Algebra, Definition 10.153.1 applies directly. $\square$

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