Chebyshev Action on Finite Fields

TL;DR

This paper explores the structure of graphs generated by Chebyshev polynomials over finite fields, revealing their symmetry and applications in prime decomposition within field extensions.

Contribution

It provides a complete description of Chebyshev polynomial graphs over finite fields and demonstrates their use in analyzing prime decomposition in field extensions.

Findings

Graphs exhibit high symmetry when f is a Chebyshev polynomial of prime degree

Complete characterization of these graphs is achieved

Applications to prime decomposition in certain field extensions are illustrated

Abstract

Given a polynomial f and a finite field F one can construct a directed graph where the vertices are the values in the finite field, and emanating from each vertex is an edge joining the vertex to its image under f. When f is a Chebyshev polynomial of prime degree, the graphs display an unusual degree of symmetry. In this paper we provide a complete description of these graphs, and also provide some examples of how these graphs can be used to determine the decomposition of primes in certain field extensions.