According to p.200-202 of the book "Reciprocity Laws: From Euler to Eisenstein" by Franz Lemmermeyer, there are two proofs for the biquadratic reciprocity law in Gauss's nachlass: one is a cyclotomic proof using Gauss sums which is very similar to Eisenstein's proof from 1844, and the other is an original contribution of Gauss - a sketch of a geometric proof of this theorem.

As for Gauss's cyclotomic proof - according to Bachmann (the editor of Gauss's number theoretical works) it cannot be dated, so it's unclear if it's original with Gauss or based on Eisenstein's proof. Since Lemmermeyer states that many of the proofs published after 1850 are just variations on Eisenstein's proof , i'm more interested in Gauss's geometric proof .


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Fortunately (and surprisingly) i found the answer very quickly - at the website of Springer they allow the readers to see the first 2 pages of each chapter, and to see the complete list of references for this book. Since the desired reference is reference 279, the relevant pages of Gauss's Nachlass are p.56-57 of volume X,1, which are entitled "Hauptmomente des Beweises fur die biquadratischen Reste". It depicts two interesting geometric figures whose meaning and connection to biquadratic reciprocity is unclear to me (by the way, there is a similar geometric fragment by Gauss which deals with "cubic residues"). Bachmann comments on it and says it's connected with the second chapter of the unpublished treatise "preconceptions to the general theory of biquadratic residues" (Gauss's werke, volume II,p.326-331).

According to Lemmermeyer, Gauss's geometric proof was reconstructed by the mathematician B. Spies in his dissertation "Der Gauss'sche Beweis des biquadratischen Reziprozitiitsgesetzes" (T.U. Braunschweig, 1983).

So i found the reference, but if anyone still wants to add usefull comments or information, i'll be glad to hear it!


  • I found a paper by Richard Hill, entitled "A GEOMETRIC PROOF OF A RECIPROCITY LAW" which states that geometric methods were applied by Gauss to reciprocity laws for the cases $K = Q$ and $K = Q(\sqrt{-1})$ (i.e, quadratic reciprocity and biquadratic reciprocity), and in addition this paper references the same long treatise by Gauss which i mentioned before ("preconceptions to the general theory of biquadratic residues",Gauss's werke, volume II, p.313-385). This shows that there is really an interesting point here on Gauss's geometric treatment of biquadratic residues, a point which was somehow neglected by mainstream historians of mathematics.

  • After a long and thorough searching on the internet, i found a most valuable source on these matters - it's the article "A foundation of class field theory applying properties of spatial figures" by Tomio Kubota, a famous researcher of number theory. On p. 14-15 he writes:

... Gauss didn't publish any proof of biquadratic reciprocity law, but one of his posthumous manuscripts did include a proof that was based on his own idea, which is rather similar to that given in that article and is actually geometric but is not sufficiently generalizable because it requires exact counts of lattice points in spatial figures.


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