Splitting fields and periods of Fibonacci sequences modulo primes

TL;DR

This paper presents an accessible algebraic approach to understanding the periods of Fibonacci sequences modulo primes, extending to general recurrences and illustrating applications of splitting fields suitable for undergraduate teaching.

Contribution

It introduces a new, algebraic method for analyzing Fibonacci periods modulo primes, avoiding complex case-by-case analysis and highlighting applications of splitting fields.

Findings

Provides a unified algebraic framework for Fibonacci periods

Extends methods to general recurrence sequences

Highlights educational applications in undergraduate courses

Abstract

What is the period of the Fibonacci sequence modulo a prime? The purpose of our brief expository paper is to illustrate an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra (rather than the case-by-case analysis found in many papers on the subject, or techniques from graduate number theory). Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in undergraduate course in abstract algebra or Galois theory.