Galois theory for finite fields

TL;DR

This note introduces Galois theory for finite fields, detailing the structure of field extensions, Galois groups, and the Galois correspondence, serving as a course handout for teaching purposes.

Contribution

It provides a clear description of Galois groups and the correspondence for finite fields, with some properties presented without proofs for educational use.

Findings

Galois group of a finite field extension is cyclic of order n.

Intermediate fields correspond to divisors of the extension degree.

Galois correspondence is established for finite field extensions.

Abstract

This note presents Galois theory for finite fields. It was written as a handout for the MAT401 course ``Polynomial equations and fields'' taught at the University of Toronto in Spring 2026. We use without proofs some basic properties of finite fields and of finite field extensions which we already covered in class. Firstly, we describe an extension $K\subset F$ of a finite field $K$ of a given degree $n$. We show that the set of all intermediate fields for this extension is in one-to-one correspondence with the set of all divisors $k$ of the degree $n$. Then we describe the Galois group of this extension which is the cyclic group of order $n$. The set of subgroups of this group also is in one-to-one correspondence with the set of all divisors $k$ of the degree $n$. It allows us to prove the Galois correspondence for that extension. In the last section, we state basic theorems of Galois theory for arbitrary fields which will be proven later in the course.