Elements of order 2 in the absolute Galois group

Solution 1:

It is a theorem of E. Artin that any non-trivial finite order element of $\mathbb Q$ is a complex conjugation. This follows from the theory of real closed fields; see here for example. (If $H \subset G$ is a non-trivial finite subgroup, then $F:=\overline{\mathbb Q}^H$ is a field whose algebraic closure is a proper finite exension; thus $F$ satisfies property 4 from the linked page. Property 5 then shows that $\overline{\mathbb Q} = F(\sqrt{-1})$, from which one sees that $H$ has order $2$; a little more argument shows that the non-trivial element of $H$ acts as a complex conjugation. Any treatment of real closed fields that is more detailed than the wikipedia page should give the details.)