Lemma 15.112.4. Let $A$ be a discrete valuation ring with fraction field $K$ and residue field $\kappa $. Let $L/K$ be a finite Galois extension with Galois group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $\mathfrak m$ be a maximal ideal of $B$. Then

the field extension $\kappa (\mathfrak m)/\kappa $ is normal, and

$D \to \text{Aut}(\kappa (\mathfrak m)/\kappa )$ is surjective.

If for some (equivalently all) maximal ideal(s) $\mathfrak m \subset B$ the field extension $\kappa (\mathfrak m)/\kappa $ is separable, then

$\kappa (\mathfrak m)/\kappa $ is Galois, and

$D \to \text{Gal}(\kappa (\mathfrak m)/\kappa )$ is surjective.

Here $D \subset G$ is the decomposition group of $\mathfrak m$.

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