number of galois extensions of local fields of fixed degree

I am sorry if I see this question only now, but since no one gave the following answer, it seems worth posting it.

There is a general formula for the number of extensions of degree $d$ of a $p$-adic field $K$ contained inside a fixed algebraic closure $\overline{K}$, which is given by $$ \# \{ L \mid K \subseteq L \subseteq \overline{K}, \, [L \colon K] = d \} = \sigma(h) \cdot \sum_{j = 0}^m \frac{p^{m+j+1} - p^{2j}}{p - 1} \cdot (p^{\varepsilon_p(j) \cdot d \cdot d_0} - p^{\varepsilon_p(j - 1) \cdot d \cdot d_0}) $$ where:

• $\sigma$ denotes the sum of divisors function;
• $h, m \in \mathbb{N}$ are the unique natural numbers such that $p \nmid h$ and $d = h \cdot p^m$;
• $\varepsilon_p(j) := \sum_{k=1}^j p^{-k}$ if $j \geq 1$, $\varepsilon_p(0) := 0$ and $\varepsilon_p(-1) := -\infty$, i.e. $p^{\varepsilon_p(-1) \cdot d} = 0$. In particular, $p^{\varepsilon_p(j) \cdot d} \in \mathbb{N}$ if $-1 \leq j \leq m$;
• $d_0 := [K \colon \mathbb{Q}_p]$.

This formula is due to Krasner, and has been proved in the paper "Nombre des extensions d'un degré donné d'un corps $\mathfrak{p}$-adique". The proof uses the same analytic techniques that go into the proof of the (much more famous) Krasner lemma.

Observe that this number is different from the number of $K$-isomorphism classes of extensions of $K$ having a given degree. This is of course due to the presence of non-Galois extensions. Here are two examples of this phenomenon:

• if $q \neq p$ is a prime then there are $q + 1$ fields $K \subseteq L \subseteq \overline{K}$ having degree $[L \colon K] = q$, but there are only two isomorphism classes of these fields: one containing the only unramified extension, and the other containing the tamely and totally ramified extensions;
• if $p \geq 3$ there are $1 + p + (p^2 - p) \cdot p$ extensions $\mathbb{Q}_p \subseteq L \subseteq \overline{\mathbb{Q}_p}$ such that $[L \colon \mathbb{Q}_p] = p$, but they form $1 + p + p^2 - p = p^2 + 1$ isomorphism classes. $p + 1$ of these contain a unique extension (which is Galois over $\mathbb{Q}_p$) and every other isomorphism class contains $p$ extensions (see for example Proposition 2.3.1 in the paper "A database of local fields" by Jones and Roberts).

Finally, let me remark that this formula is related to Serre's "mass formula", which is valid in any characteristic. This formula says that a certain "count" of totally ramified extensions of a local, non-Archimedean field $K$ of degree $d$ is equal to $d$. More precisely, $$ \sum_{L \in \Sigma_d} (\# \kappa)^{d - 1 - \mathrm{v}_K(\mathrm{disc}(L/K))} = d $$ where $\Sigma_d$ denotes the set of totally ramified extensions of $K$ which have degree $d$, and $\kappa$ is the residue field of $K$. Observe that if $p \nmid d$ then the formula can be written simply as $\# \Sigma_d = d$. Two useful references for this are:

• Serre's original paper "Une 'formule de masse' pour les extensions totalement ramifiées de degré donné d'un corps local";
• Krasner's paper "Remarques au sujet d'une note de J.-P. Serre...", in which he reproves the formula using his methods.

A Galois extension of degree $p$ has Galois group $\mathbb{Z}/p\mathbb{Z}$, so you are asking about abelian extensions of your local field. Thus the answer to your question can be obtained explicitly via local class field theory -- you can get the needed results out of Serre's Local Fields or many other books.

If $K$ contains the $p$-th roots of unity, then Kummer theory tells us that the degree $p$ Galois extensions of $K$ are in bijective correspondence with the subgroups of $K^{\times}/(K^{\times})^p$ of order $p$. The structure of $K^{\times}$ is well-known; see http://en.wikipedia.org/wiki/Local_fields or any decent book on local fields. So you can work out the answer in this case.

If $K$ doesn't contain the $p$-th roots of unity, then it becomes harder. Here are some special cases. For every $d \in \mathbb{N}$, $K$ has a unique unramified extension of degree $d$, which is necessarily cyclic - see Corollary 4.4 of these nice notes: http://websites.math.leidenuniv.nl/algebra/localfields.pdf So in particular there is a unique unramified degree p Galois extension of $K$.

If $l \neq p$, then any ramified extension of $K$ must be totally and tamely ramified. But then by 5.3 and 5.4 of the above notes, $\mathbb{Z}/p\mathbb{Z}$ must embed into the unit group of the residue field of $K$. Then by Hensel's Lemma, $K$ must contain $p$-th roots of unity and so we are reduced to the Kummer case above.

So we are left with the case $l=p$ and $K$ not containing $p$-th roots of unity. I'll think about this some more, but you should be able to use class field theory as mentioned above.