Countability of Gal(C/Q) and isomorphism of its subgroups

Question

Firstly, I want to know how can I prove that Gal(C/Q) is an uncountable group. Secondly how to show that the subgroup {g ϵ Gal(C/Q)| g continous} is isomorphic to Z/2. Really I don't know how to start so any clues are useful. The only time when I work with this type of Galois group was when I tried to show that Gal(R/Q) is trivial using R-homomorphisms and automorphisms.

Thank you in advance.

Answer 1

Hints:

1. Show that $\operatorname{Gal}(\overline{\Bbb Q}/\Bbb Q)$ is uncountable.
2. Show that any continuous element fixes $\Bbb R$.