The origins of the cubic and biquadratic reciprocity laws

Abstract

Every student of number theory is familiar with the use of the quadratic reciprocity law in deciding whether quadratic congruences have integral solutions. However, very few mathematicians are aware of the existence of cubic and biquadratic reciprocity laws which can be applied to determine whether specific congruences of third and fourth degree possess integral solutions. The absence of references to these laws in modern literature l is surprising for several reasons. First of all, they are quite similar to the quadratic reciprocity law which has continued to generate great enthusiasm since its discovery. In fact, Carl Gauss' motivation for seeking new proofs of the quadratic reciprocity law was to find methods applicable to the theory of cubic and biquadratic residues.2 Furthermore, the cubic and biquadratic reciprocity laws were so highly esteemed in the nineteenth century that a bitter dispute arose between Ferdinand Eisenstein and Carl Jacobi concerning priority of proofs. The controversy will be described in this paper following an explanation of the terms used in the statement of the reciprocity laws.