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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.

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    $\begingroup$ This seems to be an old conjecture of Shafarevich. This much is mentioned in Barry Mazur's article in "Modular Forms and Fermat's Last Theorem." $\endgroup$
    – Vesselin Dimitrov
    Commented Sep 24, 2012 at 19:11

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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 $S'=S\cap\mathrm{Ram}(K/\mathbb{Q})$...)

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  • $\begingroup$ For the second edition, it's on page 623. $\endgroup$
    – Kevin Ventullo
    Commented Nov 6, 2012 at 5:53
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    $\begingroup$ WOW! Ninety pages more. I hope they are not all taken by the preface to the second edition... $\endgroup$
    – Filippo Alberto Edoardo
    Commented Nov 6, 2012 at 9:53
  • $\begingroup$ Surprising. I thought that it was widely believed that those groups were finitely generated. $\endgroup$
    – Joël
    Commented Nov 6, 2012 at 15:11
  • $\begingroup$ One (secondary) question remains: was the finite generation really a conjecture of Shafarevich (and then, where), or is Mazur's saying so in "modular forms and Fermat's last theorem" a mistake ? $\endgroup$
    – Joël
    Commented Nov 6, 2012 at 15:13
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    $\begingroup$ FAE: on page 246 of the Deformation Theory paper (1997) that VD and Joël cited, Mazur says: "It is not even known whether or not $G_{K,S}$ is finitely generated as a topological group (although this has been conjectured by Shafarevich about thirty years ago)." $\endgroup$
    – Ben Wieland
    Commented Nov 6, 2012 at 16:37

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