Maximal real subfield of a number field Let $L\subset \mathbb{C}$ be a number field such that $L / \mathbb{Q}$ is a Galois extension, then is it true that $[L:L\cap \mathbb{R}]\leq 2$?
Thanks very much!

Yes. If $c$ is complex conjugation, $c$ acts on $L$, as an automorphism of order $1$ or $2$, so its fixed field, $L\cap\Bbb R$, is a subfield of $L$ of index $1$ or $2$.
