The Stacks project

Lemma 15.111.3. Let $A \subset B \subset C$ be extensions of discrete valuation rings. Then the ramification indices of $B/A$ and $C/B$ multiply to give the ramification index of $C/A$. In a formula $e_{C/A} = e_{B/A} e_{C/B}$. Similarly for the residual degrees in case they are finite.

Proof. This is immediate from the definitions and Fields, Lemma 9.7.7. $\square$

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