- #1

Mystic998

- 206

- 0

## Homework Statement

Okay, I'm trying to explicitly determine the Galois group of [itex]x^p - 2[/itex], 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 [tex]\textbf{Q} \subset \textbf{Q}(\zeta) \subset \textbf{Q}(\zeta,\sqrt[p]{2})[/tex] and [tex]\textbf{Q} \subset \textbf{Q}(\zeta^{n}\sqrt[p]{2}) \subset \textbf{Q}(\zeta,\sqrt[p]{2})[/tex], where [itex]0 \leq n \leq p-1[/itex], and [itex]\zeta[/itex] 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.