$\mathbb{Z}/p\mathbb{Z}$ extension of a local field

Question

Let $K/\mathbb{Q}_l$ be a finite extension. How can one prove that the number of extensions of $L/K$ such that $Gal(L/K) \cong \mathbb{Z}/p\mathbb{Z}$ is finite.

If i'm not mistaken class field theory tells us that those extensions are in bijection with the subgroups of index $p$ of $K^\times$. I want to say that such subgroups are themselves in bijections with the subgroups of index $p$ of $K^\times/(K^\times)^p$ and that since this set is finite (is it really finite ?) the result follows.

But I think that this argument is false since the following answer on mathoverflow : https://mathoverflow.net/questions/68615/number-of-galois-extensions-of-local-fields-of-fixed-degree seems to indicate that the the task of finding the exact number of such extensions is quite hard.

So I am wondering if there is an easy way to only prove the finiteness (which is all I need).

Answer 1

Since you write $Gal(L/K)\cong \mathbf Z/p\mathbf Z$, I guess that you assume $L/K$ to be normal. To count the number of all such extensions, it is convenient to introduce their compositum, say $M$, which is the maximal abelian extension of $K$ of exponent $p$ (this means that $Gal(M/K)$ is abelian, killed by $p$). By local class field theory, $Gal(M/K) \cong K^*/K^{*p}$. The latter group can be considered as a vector space over the field $\mathbf Z/p\mathbf Z$ , and we have just to compute its dimension $d$: the number we look for will be $p^d -1/p-1$. The most efficient way is to use the technique of Herbrand quotients as in Serre's "Corps Locaux", chapter 13, §4, exercise 3. Denote by $l$ the caracteristic of the residue field of $K, n=[K:\mathbf Q_l], \epsilon (K)= 1$ or $n$ according as $ l\neq p$ or $l=p$, $\delta (K) = 1 $ or $0$ according as $K$ contains a primitive root of unity or not. Then $d = 1+\epsilon(K) + \delta(K)$.

NB:The answers given in your reference to mathoverflow look complicated because, first they do not really use CFT, second they do not use the Herbrand quotient.

Answer 2

$K$ is an $\ell$-adic field, and $L$ is a cyclic, hence abelian extension of $K$. So local class field theory gives an explicit answer (yes, the group $K^×/(K^×)^p$ is finite, since $char(K)=0$ does not divide $p$). But for the above correspondence via Kummer theory we need that $K$ contains the $p$-th roots of unity. For details see "Local discriminants, kummerian extensions, and elliptic curves" by C. S. Dalawat.

In general, every $\ell$-adic field has only finitely many extensions of a given finite degree. Krasner gave formulae for the number of extensions of a given degree and discriminant in $1966$ (see Serre's book on local fields).