The Fundamental Theorem of Finite Fields: A Proof from First Principles

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.”

MSC:

• Primary 11T06

ACKNOWLEDGEMENTS

The authors would like to thank Scott Chapman, Lily Silverstein, and Wencin Poh for numerous helpful conversations. We are grateful to Jesús A. De Loera for sharing his lecture notes from the course that inspired this manuscript. We also wish to thank the referees for their insightful comments.

Additional information

Notes on contributors

Anastasia Chavez

ANASTASIA CHAVEZ received her Ph.D. in mathematics from the University of California, Berkeley. She held a visiting position at the University of California, Davis before joining Saint Mary’s College of California.

Christopher O’Neill

CHRISTOPHER O’NEILL is an Associate Professor at San Diego State University. He received his Ph.D, from Duke University and held visting positions at Texas A&M University and University of California Davis.