TITLE: Are Galois groups of Q with restricted ramification supposed to be finitely generated?
QUESTION [12 upvotes]: Fix a finite set $S$ of places of $\mathbb Q$. Let $G_{\mathbb Q,S}$ be the Galois group of the maximal extension of $\mathbb Q$ unramified outside S$. I believe that it is an open question whether this group is topologically finitely generated, that is, contains a dense, finitely generated subgroup. 

Is there a standard conjecture about whether $G_{\mathbb Q,S}$ should be finitely generated or not?
  Has anyone published an opinion?

JSE mentions a stronger conjecture: $|S|$ generators suffice; further, the existence of a special set of generators labeled by $S$ and inert at the labeling place. Is this refinement standard? (Note that a finite extension would suffice to disprove this hypothesis.)
Weakening the hypothesis, we could restrict the ramification at $S$ to be tame.

REPLY [5 votes]: I think you can look at page 532 of (the first version of) J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, Springer, 1999. They explicitly write that we do not even know precisely what to conjecture for arbitrary number field. They also add "many mathematicians (including the authors) tend to think of $G_S$ as being not finitely generated". In their notation, $G_S$ is the Galois group of the maximal unramified outside of $S$ extension of a number field $K$ which has been fixed once and for all (and $S$ is finite). I do not know if the special case $K=\mathbb{Q}$ is somehow different, but I doubt it (after all, their $G_S$ is a subgroup of your $G_{\mathbb{Q},S'}$ for $S'=S\cap\mathrm{Ram}(K/\mathbb{Q})$...)