User eof - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:26:11Z http://mathoverflow.net/feeds/user/18586 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>