Another proof of Eisenstein's law of cubic reciprocity and its supplement (Q1080876)

The reviewer [Mem. Fac. Sci., Kyushu Univ., Ser. A 28, 93--99 (1974; Zbl 0335.10008)] gave a simple proof of Eisenstein's law of cubic reciprocity. In this article the authors simplify this proof, and additionally prove a complementary law to the same reciprocity law. The method used here is based upon ideas used by \textit{P. Kaplan} [J. Fac. Sci., Univ. Tokyo, Sect. I 16, 115--145 (1969; Zbl 0216.037)] to prove the laws of quadratic and biquadratic reciprocity and the third author [Math. Mag. 49, 89--90 (1976; Zbl 0317.10006)] to prove a complementary law of quadratic reciprocity.