Remark 15.111.6. Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite separable field extension. Let $B \subset L$ be the integral closure of $A$ in $L$. Picture:

By Algebra, Lemma 10.161.8 the ring extension $A \subset B$ is finite, hence $B$ is Noetherian. By Algebra, Lemma 10.112.4 the dimension of $B$ is $1$, hence $B$ is a Dedekind domain, see Algebra, Lemma 10.120.17. Let $\mathfrak m_1, \ldots , \mathfrak m_ n$ be the maximal ideals of $B$ (i.e., the primes lying over $\mathfrak m_ A$). We obtain extensions of discrete valuation rings

and hence ramification indices $e_ i$ and residue degrees $f_ i$. We have

by Algebra, Lemma 10.121.8 applied to a uniformizer in $A$. We observe that $n = 1$ if $A$ is henselian (by Algebra, Lemma 10.153.4), e.g. if $A$ is complete.

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