The Reciprocity Law from Euler to Eisenstein

Abstract

The Reciprocity Law plays a very central role in number theory. It grew out of the theory of quadratic forms. The Quadratic Reciprocity Law was first formulated by Euler and Legendre and proved by Gauss and partly by Legendre. The search for higher reciprocity laws gave rise to the introduction and study of the Gaussian integers and more generally of algebraic numbers. Analytic methods introduced by Euler and Dirichlet in connection with the study of sets of prime numbers and primes in arithmetic progressions and their generalization by Dedekind and Weber to algebraic number fields led to a general form of the reciprocity law found and proved by Artin. This Reciprocity Law of Artin which can be considered as being an abelian reciprocity law plays a central role in class field theory. It is the starting point for a search of a more general non-abelian reciprocity law, a hint to whose existence is given by Langlands’ program.