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A short proof is given of the supplement to the law of cubic reciprocity proved by Eisenstein in 1844. Let c = (-1 + V-3)/2. Let 7r = a + bo be a primary complex prime in the Eisenstein domain ZVJl so that (1) a 2 (mod 3), b 0(mod 3), say (2) a = 3m1, b= 3n, and (3) a2 ab + b = 7zi= p, where p is a rational prime 1 (mod 3) (see for example [3, Chapter 9]). The cubic residue character (./7T)3 modulo 7T is defined by 3~~~~ (4) (a/T)3 = r if > (P-)/ ( (mod ), where r = 0 1, 2 and a C Z[o] is such that a j 0 (mod 7r). The supplement to the law of cubic reciprocity proved by Eisenstein [2] in 1844 states that (5) ((1 -2 )/7T)3 = 2m We remark that 1 X is a prime factor of 3 in Z[k]. Here is a simple proof of this result (see comment in [3, p. 115])o Let (h, k)3 denote the number of solutions (r, s) of 1 + gr g3s+k (mod p), with O < r, s < (p 1)3, where g is a primitive root (mod p) such that (gl7r)= c. (If g is such that (gl7r) = co2, we can replace g by Received by the editors January 31, 1974. AMS (MOS) subject classifications (1970). Primary 1OA15; Secondary 12C20.

Note on the supplement to the law of cubic reciprocity