3
$\begingroup$

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?

$\endgroup$
1

1 Answer 1

9
$\begingroup$

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).

$\endgroup$
2
  • $\begingroup$ Whoops! I'm not quite sure why I thought this was true. $\endgroup$ May 4, 2011 at 16:29
  • $\begingroup$ 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 $F$... $\endgroup$
    – Stefano V.
    May 4, 2011 at 19:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.