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

• Mystic998
In summary, the conversation is about determining the Galois group of x^p - 2, where p is a prime. The solution involves creating extensions of \textbf{Q} and using a primitive pth root of unity. The Galois group was found to have an order of p(p-1). A suggestion was given to think about semidirect products, and the person plans to try it for p = 5.
Mystic998

## Homework Statement

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

## 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.

Hint: Think semidirect products.

## What is a Galois group of a polynomial?

The Galois group of a polynomial is a mathematical concept that represents the set of all possible symmetries or rearrangements of the roots of a polynomial equation. It tells us how the roots of a polynomial can be permuted while still maintaining the same structure.

## Why is the Galois group of a polynomial important?

The Galois group provides important information about the solvability of a polynomial equation by radicals. It also has implications in other areas of mathematics, such as algebraic geometry and number theory.

## How is the Galois group of a polynomial calculated?

The Galois group can be calculated using Galois theory, which is a branch of mathematics that studies the relationship between field extensions and their corresponding Galois groups. In general, the Galois group can be determined by finding the automorphisms of the field generated by the roots of the polynomial.

## What are some properties of the Galois group of a polynomial?

The Galois group has several important properties, including being a finite group, having a subgroup for each intermediate field extension, and being isomorphic to the group of permutations of the roots of the polynomial. It also has a close connection to the concept of solvability by radicals.

## What can the Galois group of a polynomial tell us about the polynomial itself?

The Galois group can provide information about the symmetry and structure of the polynomial. For example, if the Galois group is a cyclic group, it means that the polynomial is symmetric and can be solved by radicals. On the other hand, if the Galois group is a non-cyclic group, it means that the polynomial is not symmetric and can't be solved by radicals.

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