The Galois group of a polynomial is a fundamental concept in algebra and Galois theory. It is a group that captures the symmetry of the roots of a polynomial and describes the field extensions that are generated by those roots. In this article, we will discuss how to find the Galois group of a polynomial.

Galois Theory Overview

Before we dive into finding the Galois group of a polynomial, we first need to understand some basic concepts of Galois theory. Galois theory is a branch of abstract algebra that deals with field extensions and their corresponding groups. A field extension is a field that contains a smaller field as a subfield. For example, the field of real numbers is an extension of the field of rational numbers. A field extension is said to be a Galois extension if it is a splitting field of a polynomial over the base field.

A polynomial is said to be solvable by radicals if its roots can be expressed using only arithmetic operations and the extraction of roots. The solvability of a polynomial is closely related to the Galois group of the polynomial. Specifically, a polynomial is solvable by radicals if and only if its Galois group is a solvable group.

Finding the Galois Group of a Polynomial

To find the Galois group of a polynomial, we follow a series of steps.

Step 1: Find the Splitting Field of the Polynomial

The first step in finding the Galois group of a polynomial is to find its splitting field. The splitting field of a polynomial is the smallest field that contains all the roots of the polynomial. To find the splitting field, we first factor the polynomial into irreducible factors over its base field. For example, consider the polynomial f(x) = x^3 – 2 over the field of rational numbers Q. We can factor f(x) as (x – ∛2)(x^2 + ∛2x + ∛4). The roots of f(x) are ∛2, ∛2ω, and ∛2ω^2, where ω = (-1 + i√3)/2 is a primitive cube root of unity. The splitting field of f(x) is the smallest field that contains all three roots, which is the field Q(∛2, ω).

Step 2: Find the Degree of the Splitting Field Extension

The second step in finding the Galois group of a polynomial is to find the degree of the splitting field extension. The degree of the splitting field extension is the degree of the polynomial over its base field Q. In the example above, the degree of the polynomial f(x) is 3, and the degree of the splitting field extension Q(∛2, ω) over Q is [Q(∛2, ω) : Q] = 6.

Step 3: Find the Automorphisms of the Splitting Field Extension

The third step in finding the Galois group of a polynomial is to find the automorphisms of the splitting field extension. An automorphism is an isomorphism from a field to itself that preserves the operations of addition and multiplication. In other words, an automorphism is a bijective function that maps each element of a field to another element of the same field, and that satisfies the following conditions: (1) f(a + b) = f(a) + f(b) for all a, b in the field, and (2) f(ab) = f(a)f(b) for all a, b in the field.

To find the automorphisms of the splitting field extension, we use the fact that any automorphism of the splitting field extension Q(∛2, ω) over Q is determined by its action on the roots of the polynomial f(x). Since the roots of f(x) are ∛2, ∛2ω, and ∛2ω^2, any automorphism of Q(∛2, ω) over Q must send each of these roots to another root of f(x). There are 6 possible ways to do this, corresponding to the 6 permutations of the roots. Each of these 6 permutations defines an automorphism of Q(∛2, ω) over Q.

For example, the automorphism that fixes Q and sends ∛2 to ∛2ω and ∛2ω to ∛2ω^2 is denoted by σ, where σ(∛2) = ∛2ω and σ(∛2ω) = ∛2ω^2. This automorphism is called a cycle of length 3. There are two other cycles of length 3, and three transpositions of pairs of roots, giving a total of 6 automorphisms of Q(∛2, ω) over Q.

Step 4: Determine the Galois Group

The Galois group of a polynomial is the group of automorphisms of its splitting field extension that fix the base field. In other words, it is the group of automorphisms of Q(∛2, ω) over Q that fix Q. To determine the Galois group, we simply list the automorphisms and their compositions, and then identify the subgroup of automorphisms that fix Q.

In the example above, the Galois group of f(x) = x^3 – 2 over Q is the subgroup of the symmetric group S3 generated by the automorphisms σ and τ, where τ(∛2) = ∛2 and τ(ω) = ω^2. The Galois group is denoted by Gal(Q(∛2, ω)/Q) and is isomorphic to the dihedral group D3.

Conclusion

In conclusion, finding the Galois group of a polynomial involves several steps. First, we must find the splitting field of the polynomial, which is the smallest field that contains all the roots. Second, we must find the degree of the splitting field extension. Third, we must find the automorphisms of the splitting field extension, which are determined by their actions on the roots of the polynomial. Finally, we must determine the subgroup of automorphisms that fix the base field, which gives us the Galois group of the polynomial. The Galois group is a powerful tool for understanding the symmetries and field extensions of a polynomial, and it has important applications in algebra, number theory, and cryptography.