Finitely Many Extensions of Fixed Degree of a Local Field
In my comment, I said there was an argument from compactness, but on thinking a little more, I realized that since the set of irreducible polynomials over the ring of integers $\mathfrak o$ of your field is not closed, that approach would be a little trickier than I hoped.
Instead, let’s do it piecemeal; I’m sure there’s a quicker argument than what I give below, but I’m not recalling it now.
It’s not Class Field Theory, it’s Kummer Theory mostly. I’ll give a proof for extensions of $\Bbb Q_p$, and there will be necessary modifications for the function-field case. I’ll show that there are only finitely many Galois extensions of degree $n$; the general case follows from this. First, notice that there is only one unramified extension of degreee $n$, so I can concentrate on totally ramified extensions.
In case we’re talking about degree $n$ prime to $p$, then Ramification Theory says that the Galois group is injected into $\kappa^*$, where $\kappa$ is the residue field; in particular, the Galois group is cyclic. Now extend the base from your original $p$-adic field $k$ to the field $k'$ you get by adjoining the $n$-th roots of unity. Now you’re talking about a cyclic extension of degree $n'$ dividing $n$, and these are in one-to-one correspondence with the subgroups of ${k'}^*/({k'}^*)^{n'}$, only finitely many ’cause the $n$-th power map is open (here is one of the places where the fact that the field is local gets used).
Now for the case that the extension is of degree $p^m$ for some $m$. Just as before, I’ll pass to the situation where the base field $k$ has $p$-th roots of unity. You have a $p$-group, so a composition series where each factor is $C_p$, cyclic of order $p$. Kummer theory again, the number of extensions of degree $p$ is finite, each of these has only finitely many extensions of degree $p$, etc. Finitely many in all.
Those are the arguments for the separate layers. I only need to say that the maximal tamely ramified extension in your general extension of degree $n$ is Galois over the base, and it all works out.
Maybe somebody else has a slicker proof; I would welcome it.
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. ¤