Let $M$ be a subfield of Complex field such that $M/\Bbb Q$ is a finite Galois extension. Show that if $[M:\Bbb Q]$ is an odd number, then $M$ is a subfield of Real field.
My current thought is since $M/\Bbb Q$ is Galois, it is a normal extension. That is, if $f(x)$ has a root in $M$, all roots of $f(x)$ are in $M$. However, I don't know how to continue from here.