This is Problem 1.7 from Gouvea's lecture notes on deformations of Galois representations. In particular, he asks you to show that it has many subgroups of finite index which are not closed. So here's what I've got so far, which may be wrong.
I can write the compositum as F = Q[√–1, √2, √3, √5, √7 ...] (can I?) and then the Galois group G = Gal(F/Q) is isomorphic to a direct product Πp (Z/2Z) where the product is taken over all primes p, as well as p=-1, and the pth component is generated by the conjugation σp defined by √p -> –√p.
An example of a subgroup which isn't closed would be the subgroup H consisting of finite products of conjugations, since for example, H contains the sequence σ2, σ2σ3, σ2σ3σ5, σ2σ3σ5σ7... which converges to the automorphism "conjugate everything", and this automorphism is not contained in H.
However this subgroup is nowhere near being finite index--it has the cardinality of the natural numbers, whereas G has the cardinality of the reals. The only finite index subgroups I can think of take are of the form Gal(F/K) where K is a finite extension of Q, but of course these are by definition all closed. So I guess I've stuffed up somewhere, and I'd be really grateful for any help?! In know this may seem a bit "homework questiony" but it's not, it's just something that's really bugging me!