Homework problem: If the field F contains a primitive nth root of unity, prove that the Galois group of $x^n - a$, for $a \in F$, is abelian.
I'm not really sure where to start here and I'm confused about the wording of the problem. I would appreciate hints.
Here are a few steps to get you started. Let $\omega$ be a primitive $n$th root of unity in $F$, and let $u$ satisfy $u^n=a$. The roots of $x^n-a$ are then $\omega^iu$ for $i=0,\ldots,n-1$. The Galois group $G$ of $x^n-a$ permutes these roots, and any element of $G$ is completely defined by where it sends $u$ (which can be chosen arbitrarily). That is, for any $i=0,\ldots,n-1$ there is a field automorphism $\sigma_i$ which maps $u$ to $\omega^iu$. See if you can take it from here.