TITLE: Finitely Many Extensions of Fixed Degree of a Local Field
QUESTION [10 upvotes]: How does one show that there are only finitely many degree $n$ extensions of a local field? I understand how this follows from class field theory in the Abelian case but don't understand how to do the non-Abelian case.

REPLY [3 votes]: About questions I have seen on the "explicit" number of extensions of fixed degree of a local field k : Krasner has shown complete (and ugly) formulae in the book "Tendances géométriques en Algèbre et Théorie des Nombres", ed. CNRS, Paris 1966, 143-169 ; a much more elegant "mass formula" has been given by Serre, Comptes Rendus Acad. Sciences Paris, 286 (1976), 1031-1036 .
As for the same problem for Galois extensions with a given Galois group G, the answer is also known when G is a p-group: if k does not contain the group of p-th roots of 1, see Safarevic, Amer. Math. Soc. Transl., 4 (1956), 59-72 ; if k does, see Yamagishi, Proc. Amer. Math. Soc., 123 (1995), 2373-2380.   ¤