MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Finite topological generation of Galois groups

Let $F/\mathbf{Q}$ be an extension of finite degree, and let $S$ be a finite set of places of $F$. Let $F_S/F$ be the maximal extension unramified outside $S$; what is the most natural way to see that $\mathrm{Gal}(F_S/F)$ is topologically finitely generated?

-
Variations of this question have been coming up once a week lately -- why? mathoverflow.net/questions/63029/… mathoverflow.net/questions/63094/… – JSE May 4 2011 at 17:15

As far as I know, in general it is an open problem to establish whether $\mathrm{Gal}(F_S/F)$ is topologically finitely generated. For example, this question is posed as a conjecture (attributed to Shafarevich) in these notes of Chenevier (see Conjecture 1.7).
 Whoops! I'm not quite sure why I thought this was true. – David Hansen May 4 2011 at 16:29 To be honest, a few minutes after writing my answer I wondered if I had misinterpreted your question and you were actually asking for the most natural strategy to attack the problem for a GIVEN number field $F$ and a GIVEN finite set $S$ of places of $F$... – Stefano V. May 4 2011 at 19:02