Is the subextension of a cyclotomic extension Galois?

Question

If $\alpha$ is a root of the polynomial $f(X)=X^p-a$ over field $F$ of characteristic not $p$, where $p$ is a prime and $a$ is not a $p$-th power. Assume that $F(\alpha) \subset F(\zeta_p)$, do we have $F(\alpha)/F$ is Galois?

Answer 1

@DonAntonio essentially answered this in his comment. The extension $F(\zeta_p)/F$ is Galois with abelian Galois group, hence every intermediate subfield $E$ (i.e. field $E$ with $F\subset E \subset F(\zeta_p)$) is Galois over F by the Fundamental Theorem of Galois Theory. Here we've used the fact that every subgroup of an abelian group is normal.