Enumerating non-abelian extensions of $\mathbb{Q}_p$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:57:23Z http://mathoverflow.net/feeds/question/78328 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78328/enumerating-non-abelian-extensions-of-mathbbq-p Enumerating non-abelian extensions of $\mathbb{Q}_p$? eof 2011-10-17T11:40:10Z 2011-10-17T11:59:17Z <p>Berkeley's collection of past qualifying exam questions contains the following:</p> <p>''What are possible extensions of degree $3$ of $\mathbb{Q}_2$?''</p> <p>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.</p> <p>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.</p> <p>I'm wondering if there's any easy ''right'' way to solve problems like this?</p> http://mathoverflow.net/questions/78328/enumerating-non-abelian-extensions-of-mathbbq-p/78329#78329 Answer by Laurent Berger for Enumerating non-abelian extensions of $\mathbb{Q}_p$? Laurent Berger 2011-10-17T11:59:17Z 2011-10-17T11:59:17Z <p>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.</p> <p>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 $p \nmid n$.</p> <ol> <li><p>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 $p=2$.</p></li> <li><p>Montrer que l'application $x \mapsto x^n$ de $1+M_K$ dans lui-même est surjective.</p></li> <li><p>Montrer que dans $O_K$, on peut écrire $\pi_K^n = p w (1+z)$ où $w^{p-1} = 1$ et $z \in M_K$.</p></li> <li><p>En déduire que $Q_p$ admet exactement $n$ extensions totalement ramifiées de degré $n$.</p></li> </ol>