I am given that $F$ is a subfield of $\mathbb R$, and $K=F(\sqrt[n]{a})$, where $a\in F$ is such that it has a real $n$th root. I want to show that , if $L$ is a Galois extension of $F$ contained in $K$, then its degree over $F$ is at most $2$.
Now, I'm guessing this has something to do with the fact that the $d$th roots of unity are real only when $d=1$ or $2$, but of course $L$ can be a field other than the splitting field of $x^d-a$. So I'm assuming I have to show that $L$ must contain the $[L:F]th$ roots of unity, but I'm not sure how to proceed. Could I have some hints?