The fundamental theorem of finite fields: a proof from first principles

TL;DR

This paper presents an accessible proof of the fundamental theorem of finite fields, designed for students without prior Galois theory knowledge, emphasizing algebraic first principles.

Contribution

It offers a novel, elementary proof of the FTFF that makes the theorem more approachable for early STEM students.

Findings

Provides a clear, algebraic proof from first principles

Enhances understanding of finite fields for students without Galois theory background

Facilitates early learning in coding theory applications

Abstract

A mathematics student's first introduction to the fundamental theorem of finite fields (FTFF) often occurs in an advanced abstract algebra course and invokes the power of Galois theory to prove it. Yet the combinatorial and algebraic coding theory applications of finite fields can show up early on for students in STEM. To make the FTFF more accessible to students lacking exposure to Galois theory, we provide a proof from algebraic "first principles."