Let $F$ be a subfield of $\mathbb R$, and let $a\in F$ be such that $\sqrt[n]{a}$ is real, and consider the extension $K=F(\sqrt[n]{a})$. I want to show that if $L$ is a Galois extension of $F$ contained in $K$, then it's a degree $1$ or $2$ extension.
I see that $K$ is also a subfield of $\mathbb R$. My "intuition" says that such a Galois extension $L$ of degree $d$ must contain the $d$th roots of unity, which are only real for $d=1, 2$, but I'm not sure how to translate it into a proof, since $L$ need not necessarily contain a $d$th root of an element of $F$.