I'm trying to solve the following problem.
Let $E = F(\alpha)$ with $\alpha^n \in F$. Assume that char($F$) does not divide $n$ and that GCD($n$, $[E : F]$) = $1$. Show that $E$ is Galois over $F$ and that $Gal(E/F)$ is abelian.
My first intuition was to show that $E$ is the splitting field over $F$ of a separable polynomial, and so I considered the polynomial $x^n - \alpha^n$ over F. But for $E$ to be the splitting field over $F$ of this polynomial, it would need to contain all $n$th roots of unity, right? And I'm not sure whether that is true. Any hints would be appreciated!