Lemma 15.114.6. Let $A$ be a discrete valuation ring with fraction field $K$. If $M/L/K$ are finite separable extensions and $M$ is tamely ramified with respect to $A$, then $L$ is tamely ramified with respect to $A$.

Proof. We will use the results of the discussion in Remark 15.111.6 without further mention. Let $C/B/A$ be the integral closures of $A$ in $M/L/K$. Since $C$ is a finite ring extension of $B$, we see that $\mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(B)$ is surjective. Hence for ever maximal ideal $\mathfrak m \subset B$ there is a maximal ideal $\mathfrak m' \subset C$ lying over $\mathfrak m$. By the multiplicativity of ramification indices (Lemma 15.111.3) and the assumption, we conclude that the ramification index of $B_\mathfrak m$ over $A$ is prime to the residue characteristic. Since $\kappa (\mathfrak m')/\kappa _ A$ is finite separable, the same is true for $\kappa (\mathfrak m)/\kappa _ A$. $\square$

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