1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Galois group of a polynomial

  1. Feb 29, 2008 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

    3. 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.
  2. jcsd
  3. Feb 29, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Hint: Think semidirect products.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook