What is the Galois group of x^p - 2 for a prime p?

[Mystic998]

Homework Statement

Okay, I'm trying to explicitly determine the Galois group of $x^p - 2$, for p a prime.

Homework Equations

The Attempt at a Solution

Okay, so what I've come up with is that I'm going to have extensions $$\textbf{Q} \subset \textbf{Q}(\zeta) \subset \textbf{Q}(\zeta,\sqrt[p]{2})$$ and $$\textbf{Q} \subset \textbf{Q}(\zeta^{n}\sqrt[p]{2}) \subset \textbf{Q}(\zeta,\sqrt[p]{2})$$, where $0 \leq n \leq p-1$, and $\zeta$ is a primitive pth root of unity. Using that information, I was able to come up with the fact that the Galois group has order p(p-1), but I can't really do much beyond that. I'm going to try figuring it out for p = 5 just to see if it's instructive, but in the meantime suggestions would be appreciated.

 

[morphism]

Hint: Think semidirect products.