Lemma 15.111.4. Let $A \subset B$ be an extension of discrete valuation rings inducing the field extension $K \subset L$. If the characteristic of $K$ is $p > 0$ and $L$ is purely inseparable over $K$, then the ramification index $e$ is a power of $p$.
Proof. Write $\pi _ A = u \pi _ B^ e$ for some $u \in B^*$. On the other hand, we have $\pi _ B^ q \in K$ for some $p$-power $q$. Write $\pi _ B^ q = v \pi _ A^ k$ for some $v \in A^*$ and $k \in \mathbf{Z}$. Then $\pi _ A^ q = u^ q \pi _ B^{qe} = u^ q v^ e \pi _ A^{ke}$. Taking valuations in $B$ we conclude that $ke = q$. $\square$
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