A Note on Powers in Finite Fields

TL;DR

This paper investigates criteria for the existence of powers in finite fields, generalizing classical results using algebraic methods suitable for educational purposes and applications in coding and cryptology.

Contribution

It extends Euler's criterion for squares to arbitrary finite fields using algebraic techniques, facilitating understanding and application in various mathematical fields.

Findings

Generalized Euler's criterion for powers in finite fields

Provided algebraic proofs suitable for teaching

Enhanced understanding of polynomial solutions in finite fields

Abstract

The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years the subject has found important applications in the modelling of problems from applied mathematical fields such as signal analysis, system theory, coding theory and cryptology. In this connection it is of interest to know criteria for the existence of squares and other powers in arbitrary finite fields. Making good use of polynomial division in polynomial rings over finite fields, we have examined a classical criterion of Euler for squares in odd prime fields, giving it a formulation which is apt for generalization to arbitrary finite fields and powers. Our proof uses algebra rather than classical number theory, which makes it convenient when presenting basic methods of applied algebra in the classroom.