The Quadratic and Cubic Characters of 2

TL;DR

This paper investigates the cubic character of 2 modulo primes, using Eisenstein integers, Gauss and Jacobi sums, and cubic reciprocity law, building on historical developments in higher reciprocity laws.

Contribution

It introduces a comprehensive approach to evaluate the cubic character of 2, integrating algebraic integers and classical sums, with historical context and new insights.

Findings

Connection between cubic character of 2 and Eisenstein integers

Application of Gauss and Jacobi sums to cubic reciprocity

Historical analysis of higher reciprocity laws

Abstract

The solvability of the cubic congruence $x^{3}\equiv 2\pmod{p}$ is referred to as the $\textit{cubic character of 2}$. In evaluating the cubic character of 2, we introduce the Eisenstein integers, Gauss and Jacobi sums, and the law of cubic reciprocity. We motivate this proof by giving ample historical information surrounding the early development of higher reciprocity laws as well as Gauss' proof of the solvability of the quadratic congruence $x^{2}\equiv 2\pmod{p}$; conventionally the $\textit{quadratic character of 2}$. We simultaneously outline other relevant contributions by Fermat, Euler, Legendre, Jacobi, and Eisenstein.