Niklas Hellberg: Finite Fields: An Introduction

Abstract: This paper presents the foundational theory of finite fields through several algebraic perspectives. Our aim is to develop a clear understanding of finite field structure and to illustrate its applications in a pedagogically accessible way. We begin with a historical overview of finite fields, followed by an introduction to the core algebraic concepts. A group-theoretic approach is then used to analyze the cyclic and symmetric properties of finite fields. We subsequently examine the Frobenius map and cyclotomic cosets, emphasizing their role in describing the internal symmetries of finite fields. The theoretical basis connecting finite fields to polynomials is introduced as a basis for computational methods, while linear-algebraic viewpoints connect finite fields to vector space structures and provide the foundation for modern linear coding theory. Finally, we discuss key applications building