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

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SUMMARY

The Galois group of the polynomial x^p - 2, where p is a prime, is determined to have an order of p(p-1). The discussion outlines the field extensions Q ⊆ Q(ζ) ⊆ Q(ζ,√[p]{2}) and Q ⊆ Q(ζ^n√[p]{2}) ⊆ Q(ζ,√[p]{2}), where ζ is a primitive pth root of unity. The analysis suggests that understanding semidirect products is crucial for further exploration of the Galois group structure, particularly for specific primes like p = 5.

PREREQUISITES
  • Understanding of Galois theory and field extensions
  • Familiarity with primitive roots of unity, specifically pth roots
  • Knowledge of semidirect products in group theory
  • Basic algebraic number theory concepts
NEXT STEPS
  • Explore the structure of semidirect products in Galois groups
  • Investigate the Galois group of x^5 - 2 as a specific case
  • Study the implications of the order of Galois groups on field extensions
  • Learn about the application of cyclotomic fields in Galois theory
USEFUL FOR

Mathematicians, particularly those specializing in algebra, number theory, and Galois theory, will benefit from this discussion, as well as students tackling advanced topics in field theory.

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Homework Statement



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

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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.
 
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Hint: Think semidirect products.
 

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