Question

Berkeley's collection of past qualifying exam questions contains the following:

''What are possible extensions of degree $3$ of $\mathbb{Q}_2$?''

I'm trying to figure out what the general approach is to attack a question like this. In this particular case, we know that $\mathbb{Q}_2^\times\simeq \mathbb{Z}\times \mathbb{Z}_2^\times$, where $\mathbb{Z}_2^\times$ is a pro-$2$ group. It follows that there is only one abelian extension of degree $3$ which would be the unramified one. Hence, all other such extensions are totally ramified.

Thus we are left with enumerating the totally ramified extensions. Here, the only approaches I can come up with is using the idea that such extensions are given by roots of Eisenstein polynomials. The standard proof that $\mathbb{Q}_p$ has a finite number of extensions of a particular degree, then shows that such polynomials are in bijection with a compact space and then uses Krasner's lemma to find a finite cover of this such that all the polynomials in the subsets of the cover have the same splitting fields. However, I can't really get anywhere applying this, as it seems to give duplicates.

I'm wondering if there's any easy ''right'' way to solve problems like this?

Answer 1

This is standard stuff. Here is (in French) the solution as an exercise, copy-pasted from the final exam of a course I gave on local fields.

Soit $K$ une extension totalement ramifiée de degré $n$ de $Q_p$ et $\pi_K$ une uniformisante de $K$. On suppose pour l'instant que $p \nmid n$.

1. Montrer que si $w \in Q_p$ et $w^n=1$, alors $w^m=1$ où $m = n \wedge (p-1)$ (si $p \neq 2$) et $m=2$ si $p=2$.

2. Montrer que l'application $x \mapsto x^n$ de $1+M_K$ dans lui-même est surjective.

3. Montrer que dans $O_K$, on peut écrire $\pi_K^n = p w (1+z)$ où $w^{p-1} = 1$ et $z \in M_K$.

4. En déduire que $Q_p$ admet exactement $n$ extensions totalement ramifiées de degré $n$.