The Galois Group and Representation Theory - Time Travel, Quantum Entanglement and Quantum Computing

Galois Group

Overview

The Galois group is a concept from the field of algebra, specifically in the study of field theory and polynomial equations. It is named after the French mathematician Évariste Galois.

Key Concepts

Field Extensions: A field extension $L / K$ is a pair of fields $K$ and $L$ such that $K$ is a subfield of $L$ .
Automorphisms: An automorphism of a field $L$ is a bijective map from $L$ to itself that respects the field operations (addition and multiplication).
Galois Group: Given a field extension $L / K$ , the Galois group $Gal(L/K)\text{Gal}(L/K)$ is the group of all automorphisms of $L$ that fix every element of $K$ .

Example

Consider the field extension $Q(2)/Q\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ :

The automorphisms of $Q(2)\mathbb{Q}(\sqrt{2})$ that fix $Q\mathbb{Q}$ are the identity map and the map sending $2\sqrt{2}$ to $−2-\sqrt{2}$ .
Therefore, the Galois group $Gal(Q(2)/Q)\text{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q})$ has two elements, often denoted by ${id,σ}\{ \text{id}, \sigma \}$ , where $σ(2)=−2\sigma(\sqrt{2}) = -\sqrt{2}$ .

Representation Theory

Overview

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces.

Key Concepts

Group Representations: A representation of a group $G$ is a homomorphism from $G$ to the general linear group $GL(V)\text{GL}(V)$ of a vector space $V$ . Essentially, it maps elements of $G$ to invertible matrices in such a way that group operations are preserved.
Modules: A module over a ring is a generalization of the concept of a vector space over a field. Representation theory can also be viewed as the study of modules over group algebras.
Characters: The character of a representation is a function that associates to each group element the trace of its matrix in the representation. Characters provide a powerful tool for studying and classifying representations.

Example

Consider the symmetric group $S_3$ , which is the group of all permutations of three elements:

A simple representation of $S_3$ is the permutation representation on $R3\mathbb{R}^3$ , where each permutation $σ∈S3\sigma \in S_3$ is represented by the matrix that permutes the standard basis vectors of $R3\mathbb{R}^3$ according to $σ\sigma$ .

Importance

Representation theory has applications in many areas of mathematics and science, including number theory, geometry, and physics.
It provides a way to study groups by understanding their action on vector spaces, making complex group-theoretic problems more manageable by translating them into linear algebraic terms.

By exploring the Galois groups and representation theory, mathematicians can gain deeper insights into the structure and symmetries of algebraic objects and their interrelationships.