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

We deduce a formula enumerating the isomorphism classes of extensions of a $\text{kp}$-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 $K\mathrm{/}{\text{Q}}_p$, and $K({\zeta }_{p^m})\mathrm{/}K$ obtained adding the $p^m$-th roots of 1, for all $p^m$ dividing $e$.