Octic Reciprocity

The proof of the octic reciprocity law differs considerably from those in the cubic and quartic cases; in fact one needs methods that are much more sophisticated than Gauss sums, and this is why we resurrect elliptic Gauss sums These were introduced by Eisenstein while he was working on octic reciprocity and the division of the lemniscate; apparently, nobody ever bothered to study these sums that I wouldn’t have hesitated to call Eisenstein sums were it not for the fact that this name is already being used for the sums that we have studied in Chapter 4. In contrast to the full octic reciprocity law, rational octic reciprocity laws can be proved quite easily: the octic version of Burde’s reciprocity law is presented in Section 9.1, Eisenstein’s octic reciprocity law and the formulas of Western are discussed in Section 9.2, and the proof of Scholz’s octic reciprocity law is given in Section 9.5.