The Reciprocity Law

Abstract

Gauss called the quadratic reciprocity law “the golden theorem.” He was the first to give a valid proof of this theorem. In fact, he found nine different proofs. After this he worked on biquadratic reciprocity, obtaining the correct statement, but not finding a proof. The first to do so were Eisenstein and Jacobi. The history of the general reciprocity law is long and complicated involving the creation of a good portion of algebraic number theory and class field theory. By contrast, it is possible to formulate and prove a very general reciprocity law for A = F[T] without introducing much machinery. Dedekind proved an analogue of the quadratic reciprocity law for A in the last century. Carlitz thought he was the first to prove the general reciprocity law for F[T]. However O. Ore pointed out to him that F.K. Schmidt had already published the result, albeit in a somewhat obscure place (Erlanger Sitzungsberichte, Vol. 58–59, 1928). See Carlitz [2] for this remark and also for a number of references in which Carlitz gives different proofs the reciprocity law. We will present a particularly simple and elegant proof due to Carlitz. The only tools necessary will be a few results from the theory of finite fields.KeywordsFinite FieldResidue ClassIrreducible PolynomialCoset RepresentativeClass Field TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.