How to Determine Galois Groups and Subfields?

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In summary, The Galois group for a polynomial can be determined by finding the set of solutions and taking the permutation group of that set. The subfields of the splitting field can also be determined using this method. For polynomials of the form x^p-2, the Galois group is the permutation group on p objects.
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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.
 

FAQ: How to Determine Galois Groups and Subfields?

What is a Galois group?

A Galois group is a mathematical concept used in the field of abstract algebra to describe the symmetries of a polynomial equation. It is named after French mathematician Évariste Galois.

How do I determine the Galois group of a polynomial?

The Galois group of a polynomial can be determined by factoring the polynomial into irreducible factors and then analyzing the permutations of the roots of the polynomial. The group will be isomorphic to the subgroup of the symmetric group that describes these permutations.

What is the significance of the Galois group?

The Galois group is significant because it allows us to understand the symmetries of a polynomial equation and to determine whether the equation can be solved using radicals. It is also used in many other areas of mathematics, such as number theory and geometry.

Can the Galois group of a polynomial be infinite?

Yes, the Galois group of a polynomial can be finite or infinite. For example, the Galois group of the polynomial x^2 - 2 is infinite, while the Galois group of the polynomial x^2 - 2x + 2 is the symmetric group S2, which is finite.

Are there any shortcuts or tricks for determining the Galois group?

There is no shortcut or trick for determining the Galois group of a polynomial. It requires a thorough understanding of abstract algebra and the use of specific techniques, such as the Fundamental Theorem of Galois Theory and the Sylow theorems.

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