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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


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    Hint: Think semidirect products.
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