Let $k$ be a field with fixed separable closure $k_s$. Consider the profinite group $Gal(k_s|k)=: G$ with the corresponding topology. Furthermore, let $U \subset G$ be an open subgroup. Every subgroup of G can be naturally identified with a group of the form $Gal(k_s|L)$, w./ $L|k$ some subextension of $k$. So far so good, just the following thing buggs me:
My question: Is the extenstion $L|k$ in this particular case ($U$ open subgroup) necessarily finite? Thanks in advance.