The Quadratic and Cubic Characters of 2

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.