The Stacks project

Lemma 58.12.2. A normal local ring with separably closed fraction field is strictly henselian.

Proof. Let $(A, \mathfrak m, \kappa )$ be normal local with separably closed fraction field $K$. If $A = K$, then we are done. If not, then the residue field $\kappa $ is algebraically closed by Lemma 58.12.1 and it suffices to check that $A$ is henselian. Let $f \in A[T]$ be monic and let $a_0 \in \kappa $ be a root of multiplicity $1$ of the reduction $\overline{f} \in \kappa [T]$. Let $f = \prod f_ i$ be the factorization in $K[T]$. By Algebra, Lemma 10.38.5 we have $f_ i \in A[T]$. Thus $a_0$ is a root of $f_ i$ for some $i$. After replacing $f$ by $f_ i$ we may assume $f$ is irreducible. Then, since the derivative $f'$ cannot be zero in $A[T]$ as $a_0$ is a single root, we conclude that $f$ is linear due to the fact that $K$ is separably algebraically closed. Thus $A$ is henselian, see Algebra, Definition 10.153.1. $\square$

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