Let $F$ be a field such that char($F$) does not divide $n$, where $\alpha$ is an element of an extension field of $F$ such that $\alpha^n \in F$, and suppose also that GCD($n$, $[F(\alpha) : F]$) = 1. Then if $\zeta$ is a primitive $n$th root of unity, it follows that $F(\alpha, \zeta)$ is Galois over $F$, since it is the splitting field of the separable polynomial $x^n - \alpha^n$ over $F$.
Now I want to try showing that the Galois group of $F(\alpha, \zeta)$ over $F$ is abelian, but I'm not sure of the best way to proceed. Any help would be appreciated!