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?


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 $n$.


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