TITLE: $Aut_\mathbb{Q}(\overline{\mathbb{Q}})$ is uncountable
QUESTION [6 upvotes]: How do you show that $Aut_\mathbb{Q}(\overline{\mathbb{Q}})$ is uncountable ?
Thanks in advance

REPLY [5 votes]: In general, a profinite space is either finite or uncountable. (A profinite space is a compact, totally disconnected, Hausdorff space. The proof that such a space is either finite or uncoutable is completely topological; see if you can come up with it!)
But $G_{\mathbf Q}$, which is profinite, cannot be finite, since there are finite Galois extensions of $\mathbf Q$ of arbitrary high degree (and their Galois groups are quotient of $G_{\mathbf Q}$).