Galois Theory

TL;DR

This paper provides comprehensive undergraduate notes on Galois theory, covering field extensions, Galois groups, solvable polynomials, and finite fields, with proofs and classical constructions.

Contribution

It offers an accessible, structured presentation of Galois theory concepts, including proofs and classifications, tailored for undergraduate students.

Findings

Proof that solvable polynomials have solvable Galois groups

Classification of finite fields

Fundamental theorem of Galois theory

Abstract

These are the notes for an undergraduate course at the University of Edinburgh, 2021-2023. Assuming basic knowledge of ring theory, group theory and linear algebra, the notes lay out the theory of field extensions and their Galois groups, up to and including the fundamental theorem of Galois theory. Also included are a section on ruler and compass constructions, a proof that solvable polynomials have solvable Galois groups, and the classification of finite fields.