Let $K$ be a number field i.e. a finite extension of $\mathbb{Q}$, $\overline{K}$ a fixed separable closure of $K$, and $G\_K:=\mathrm{Gal}(\overline{K}/K)$ the absolute Galois group of $K$. Let $S$ be a finite set of non-Archimedean places of $K$, or equivalently, a finite set of non-zero prime ideals in the ring of integers $\mathcal{O}\_K$ of $K$. We define
$$\overline{K}\_S:=\text{the maximal algebraic extension of $K$ in $\overline{K}$ unramified outside $S$}$$
and
$$G\_{K,S}:=\mathrm{Gal}(\overline{K}\_S/K).$$


At the beginning of Barry Mazur's part of *Modular Forms and Fermat’s Last Theorem*, he states:



> 
> What is the "structure" of $G\_{K,S}$ - whatever that means? It is not even known whether or not $G\_{K,S}$ is finitely generated as a topological group (**although this has been conjectured to be the case by Shafarevich about thirty years ago**).
> 
> 
> 


Does anyone have a source for this conjecture? Since this book was published in 1997, I guess the conjecture that Mazur is referring to would've been formulated in the mid-60s.


The only conjectures by Shafarevich that I could find are (1) the Tate-Shafarevich conjecture on the Tate-Shafarevich group, and (2) concerning the absolute Galois group of $\mathbb{Q}^{\text{ab}}$. I don't think either is equivalent to the above, but if so, what's the intuition behind that?



Shafarevich's conjecture on Galois groups over fields ramified at finitely many places
In the footnote on Page 11 of [Fernando Q. Gouvea: Deformations of Galois Representations],



> 
> The reference is
> 
> 
> [I.R. Shafarevich, *Algebraic number fields*, Proceedings of the
> international Congress of Mathematicians, Stockholm, 1962 (Djursholm),
> Inst. Mittag-Leffler, 1963, Translated version reprinted in I.R.
> Shafarevich, *Collected Mathematical Papers* (Springer- Verlag, 1989),
> pp. 283-294, pp. 163-176.],
> 
> 
> but we should note that there Shafarevich simply asks whether it is
> the case that $G\_{K,S}$ is finitely generated for any number field
> (and whether the number of generators can be bounded in terms of the
> number of elements of $S$). His main reason for posting the question
> is that the analogous statement is true for function field over
> $\mathbb{C}$.
> 
> 
> 


