Determination of the number of isomorphism classes of extensions of a p-adic field

We deduce a formula enumerating the isomorphism classes of extensions of a p-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/Qp, and K(ζpm)/K obtained adding the pm-th roots of 1, for all pm dividing e.