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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