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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?

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Variations of this question have been coming up once a week lately -- why?…… – JSE May 4 '11 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).

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Whoops! I'm not quite sure why I thought this was true. – David Hansen May 4 '11 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 '11 at 19:02

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