The Stacks project

Lemma 15.112.1. Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension with Galois group $G$. Then $G$ acts on the ring $B$ of Remark 15.111.6 and acts transitively on the set of maximal ideals of $B$.

Proof. Observe that $A = B^ G$ as $A$ is integrally closed in $K$ and $K = L^ G$. Hence this lemma is a special case of Lemma 15.110.8. $\square$

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