Galois group of a polynomial

The Galois group of a polynomial is a concept in the field of algebra, specifically within Galois theory, which studies the relationship between field extensions and group theory. The Galois group of a polynomial provides deep insights into the solvability and structure of the roots of the polynomial.

Key Concepts

1. Polynomial and Its Roots

• A polynomial $f(x)$ with coefficients in a field $K$ (typically $\mathbb{Q}$, $\mathbb{R}$, or $\mathbb{C}$) can be written as f(x)=anxn+an−1xn−1+⋯+a0f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0f(x)=an​xn+an−1​xn−1+⋯+a0​.
• The roots of $f(x)$ are the solutions to the equation $f(x)=0$.

2. Splitting Field

• The splitting field $L$ of $f(x)$ over $K$ is the smallest field extension of $K$ that contains all the roots of $f(x)$.
• If $f(x)$ has degree $n$, the splitting field $L$ will be an extension of $K$ with a degree that divides $n!$.

3. Automorphisms

• An automorphism of a field $L$ is a bijective map from $L$ to itself that preserves addition and multiplication.
• The automorphisms of $L$ that fix $K$ form a group under composition.

4. Galois Group

• The Galois group $\text{Gal}(L/K)$ of the polynomial $f(x)$ is the group of all automorphisms of $L$ that fix every element of $K$.

Steps to Determine the Galois Group

1. Find the Roots: Determine the roots of the polynomial $f(x)$. This can be done exactly for polynomials of degree 2, 3, and 4, but for higher degrees, it might require numerical or symbolic methods.
2. Construct the Splitting Field: Identify the splitting field $L$ of the polynomial over the base field $K$.
3. Determine Automorphisms: Identify all automorphisms of the splitting field that fix the base field.
4. Form the Group: The set of these automorphisms forms the Galois group, denoted $\text{Gal}(L/K)$.

Examples

1. Quadratic Polynomial

Consider f(x)=x2−2f(x) = x^2 – 2f(x)=x2−2 over $\mathbb{Q}$:

• The roots are ±2\pm \sqrt{2}±2​.
• The splitting field is Q(2)\mathbb{Q}(\sqrt{2})Q(2​).
• The Galois group consists of two elements: the identity automorphism and the automorphism sending 2\sqrt{2}2​ to −2-\sqrt{2}−2​.
• Thus, Gal(Q(2)/Q)≅Z/2Z\text{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z}Gal(Q(2​)/Q)≅Z/2Z.

2. Cubic Polynomial

Consider f(x)=x3−2f(x) = x^3 – 2f(x)=x3−2 over $\mathbb{Q}$:

• The roots are 23,23ω,23ω2\sqrt[3]{2}, \sqrt[3]{2}\omega, \sqrt[3]{2}\omega^232​,32​ω,32​ω2, where $\omega$ is a primitive cube root of unity.
• The splitting field is Q(23,ω)\mathbb{Q}(\sqrt[3]{2}, \omega)Q(32​,ω).
• The Galois group has order 6 and is isomorphic to S3S_3S3​, the symmetric group on 3 elements.

Importance of the Galois Group

• Solvability by Radicals: A polynomial is solvable by radicals if and only if its Galois group is a solvable group.
• Field Extensions: The structure of the Galois group provides information about the intermediate fields between $K$ and $L$.
• Symmetry: The Galois group reflects the symmetries of the roots of the polynomial and helps in understanding the nature of the solutions.