TITLE: Criteria for topologically finitely generated profinite groups 
QUESTION [6 upvotes]: 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?

REPLY [5 votes]: The following profinite group is not finitely generated but has only finitely many open subgroups of each degree.
$$G=\prod_{n=5}^\infty (Alt_n)^{(n+1)!}.$$
Note that you need at least $n+1$ elements in order to generate $(Alt_n)^{(n+1)!}$.