The origins of the cubic and biquadratic reciprocity laws (Q1239148)

The cubic and biquadratic reciprocity laws are the third and fourth power analogues of the quadratic reciprocity law. Yet, although the quadratic reciprocity law is widely known and discussed in most texts on number theory, the cubic and biquadratic laws are, for the most part, unknown and unavailable in modern literature. The purpose of this article is to inform the readers of the existence of the cubic and biquadratic laws, and to give their historical background. After an explanation of the statement and usefulness of the reciprocity laws, their origins are traced from the initial studies of Carl Gauss in 1828 through the first published proofs in the 1840's. Especially well documented is the dispute between Carl Jacobi and Ferdinand Eisenstein over the originality of Eisenstein's proofs. The article concludes with a brief description of later developments which culminate in Artin's Reciprocity Law.