On the theory of cubic residues and nonresidues

Abstract

The assertion (1.3) is now called the general cubic reciprocity law ; it was first proved by G. Eisenstein. Let p be a prime of the form 3n+1. It is well known that there are unique integers L and |M | such that 4p = L+27M with L ≡ 1 (mod 3). It follows that ( L 3M )2 ≡ −3 (mod p) and therefore m(p−1)/3 ≡ 1, ( − 1 − L 3M )/2 or ( −1 + L 3M ) /2 (mod p) for any integer m 6≡ 0 (mod p). In 1827 Jacobi [J] established the following rational cubic reciprocity law.