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Determination of the number of isomorphism classes of extensions of a $\kp$-adic field

We deduce a formula enumerating the isomorphism classes of extensions of a $\kp$-adic field $K$ with given ramification $e$ and inertia $f$. The formula follows from a simple group-theoretic lemma, plus the Krasner formula and an elementary class field theory computation. It shows that the number of classes only depends on the ramification and inertia of the extensions $K/\Q_p$, and $K(ζ_{p^m})/K$ obtained adding the $p^m$-th roots of 1, for all $p^m$ dividing $e$.