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.