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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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). 

