Distributions of $n$th Powers in Finite Fields

TL;DR

This paper investigates the distribution of nth power residues in finite fields and primes, using character sums to analyze their properties and extend the understanding to arbitrary finite fields.

Contribution

It introduces a method to analyze nth power distributions in finite fields and primes, extending previous results to more general finite field settings.

Findings

Distribution of nth power residues modulo prime p characterized.

Method extended to arbitrary finite fields.

Provides new insights into power residue distributions.

Abstract

In this paper, we first find the distribution of nth power residues modulo a prime $p$ by analyzing sums involving Dirichlet characters. We then extend this method to characterize the distribution of powers in arbitrary finite fields.