Artin Reciprocity and Mersenne Primes

On March 3, 1998, the centenary of Emil Artin was celebrated at the Universiteit van Amsterdam. This paper is based on the two morning lectures, enti-tled`Artin reciprocity and quadratic reciprocity' and`Class eld theory in practice', which were delivered by the authors. It provides an elementary introduction to Artin reciprocity and illustrates its practical use by establishing a recently observed property of Mersenne primes. Emil Artin was born on March 3, 1898 in Vienna, as the son of an art dealer and an opera singer, and he died on December 20, 1962 in Hamburg. For a description of his life and his personality we refer the reader to Richard Brauer's beautifully written obituary 2]. Artin was one of the founding fathers of modern algebra. Van der Waerden acknowledged his debt to Artin and to Emmy Noether (1882{1935) on the title page of his Moderne Algebra (1930{31), which indeed was originally conceived to be jointly written with Artin. The single volume that contains Artin's collected papers, published in 1965 1], is one of the other classics of twentieth century mathematics. Artin's two greatest accomplishments are to be found in algebraic number theory. Here he introduced the Artin L-functions (1923), which are still the subject of a major open problem, and he formulated (1923) and proved (1927) Artin's reciprocity law, to which the present paper is devoted. Artin's reciprocity law is one of the cornerstones of class eld theory. This branch of algebraic number theory was during the prewar years just as forbidding to the mathematical public as modern algebraic geometry was to be in later years. It is still not the case that the essential simplicity of class eld theory is known to \any arithmetician from the street" 11]. There is indeed no royal road to class eld theory, but, as we shall show, a complete and rigorous statement of Artin's reciprocity law is not beyond the scope of a rst introduction to the subject. To illustrate its usefulness in elementary number theory, we shall apply it to prove a recently observed property of Mersenne primes.