Nick Halloway wrote: > Is it true that you could have a subgroup of finite index in the Galois > group of the algebraic numbers over the rationals that isn't the group > that fixes an intermediate field?
I doubt it. Profinite Galois groups set up a correspondence between their closed subgroups and intermediate fields. Subgroups of finite index should be open and closed. I suppose to see that it is useful to quote the lemma that inside a subgroup H of index n in G there is a normal subgroup N of index at most n! . WLOG we can assume a normal subgroup, therefore. The question reduces to whether there are discontinuous homomorphisms from these profinite groups to finite groups carrying the discrete topology. I think this is just a matter of chasing back through the definition of an inverse system.