The Stacks project

Lemma 15.111.2. Let $A \subset B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. If the extension $L/K$ is finite, then the residue field extension is finite and we have $ef \leq [L : K]$.

Proof. Finiteness of the residue field extension is Algebra, Lemma 10.119.10. The inequality follows from Algebra, Lemmas 10.119.9 and 10.52.12. $\square$

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