How to Determine Galois Groups and Subfields?

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SUMMARY

This discussion focuses on determining Galois groups and subfields for specific polynomials, particularly (x^2-2)(x^2-3)(x^2-5) and x^p-2 for prime p. The Galois group for the polynomial x^p-2 is identified as the permutation group on p objects, arising from the roots of the equation xp = a, which are expressed in terms of the principal pth root of unity. The discussion emphasizes the importance of understanding the structure of these groups and their corresponding subfields in the context of field theory.

PREREQUISITES
  • Understanding of Galois theory
  • Familiarity with polynomial roots and splitting fields
  • Knowledge of permutation groups
  • Concept of roots of unity
NEXT STEPS
  • Study the properties of Galois groups in field extensions
  • Learn about splitting fields and their subfields
  • Explore the relationship between roots of unity and Galois theory
  • Investigate specific examples of Galois groups for various polynomials
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Mathematicians, particularly those specializing in algebra and field theory, as well as students seeking to deepen their understanding of Galois groups and their applications in solving polynomial equations.

minibear
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I have trouble in determining Galois group.
Can anyone help me with the following question:

how to determine the Galois group of (x^2-2)(x^2-3)(x^2-5), determine all the subfields of the splitting field of this polynomial?

how to determine the elements of the galois group of x^p-2 for p is prime.

Thanks a lot!
 
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The set of solutions of xp= a are of the form [itex]|a|^{1/p}\omega^i[/itex] with i ranging from 0 to p-1, where [itex]\omega[/itex] is the "principal pth root of unity". The Galois group is the permutation group of that set: the permutation group on p objects.
 

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