Cubes in finite fields and related permutations

TL;DR

This paper investigates the permutation formed by cubic residues modulo primes of the form 3n+1, specifically determining its sign based on properties of primitive roots and cubic residues.

Contribution

It provides a novel analysis of the permutation sign of cubic residues modulo primes of the form 3n+1, linking number theory and permutation properties.

Findings

Determined the sign of the permutation of cubic residues modulo p.

Connected permutation sign to properties of primitive roots and cubic residues.

Enhanced understanding of structure of cubic residues in finite fields.

Abstract

Let $p=3n+1$ be a prime with $n\in\mathbb{N}=\{0,1,\cdots\}$, and let $g\in\mathbb{Z}$ be a primitive root modulo $p$. Let $0<a_1<\cdots<a_n<p$ be all the cubic residues modulo $p$ in the interval $(0,p)$. Then clearly the sequence $$a_1\ {\rm mod}\ p,\ a_2\ {\rm mod}\ p,\cdots, a_n\ {\rm mod}\ p$$ is a permutation $s_p(g)$ of the sequence $$g^3\ {\rm mod}\ p,\ g^6\ {\rm mod}\ p,\cdots, g^{3n}\ {\rm mod}\ p.$$ In this paper, we shall determine the sign of this permutation.