I'm having some difficulty with a problem. Here is the problem statement.
Let $p$ be a prime, and $G$ be the Galois group of $f(x) = x^p - 2$ over $E$ where $E = \mathbb{Q}(\xi)$ and $\xi$ a primitive $p^{th}$ root of unity. Prove that the group $G$ is isomorphic to $\mathbb{Z}_p$.
I know that if $w = \sqrt[p]{2}$ then the set of roots of $f$ is $S = \{w, w\xi, \ldots, w\xi^{p-1}\}$. Furthermore, I know that for any $\alpha \in G$ $\alpha$ can only send roots of $f$ to other roots of $f$. Therefore, my intuition is that we could maybe make an isomorphism $\phi : G \rightarrow \mathbb{Z}_p$ where for $\alpha \in G$ $\phi(\alpha) = k$ where $\alpha(w) = w\xi^k$.
If I could get a hint for this problem that would be great!