Q1: Do we have a criterion which allows us to say when is a profinite group $G$ topologically finitely generated?


For example, if $G$ is topologically finitely generated then, for a fixed integer $N$,
there are only finitely many open subgroups $H\leq G$ such that $[G:H]=N$. In general, I guess that this condition is not sufficient.


Q2: Where can I find a proof that the absolute galois of $ \mathbf{Q}\_p $ is topologically finitely generated? 



Criteria for topologically finitely generated profinite groups 
See the theorem of Mann quoted by Jaikin in [this question](https://mathoverflow.net/questions/63029/finitely-generated-galois-groups) for a nice criterion guaranteeing topological finite generation in terms of the number of open subgroups of index N. See Pete Clark's answer to that question for pointers towards verifying that criterion for Gal(Q\_p).


