Global Class-Field Theory

W. Narkiewicz

Institute of Mathematics, Wroc aw University, pl. Grunwaldzki 2/4, 50-384, Wroc aw, Poland
e-mail: narkiew@math.uni.wroc.pl

Contents

1. Introduction
2. The classical approach
3. The modern approach
4. Reciprocity laws
5. Density theorems
6. Kronecker's ``Jugendtraum'' and explicit class-fields
7. Class-field-tower problem
References

Introduction

The aim of the class-field theory is to describe all Abelian extensions of a given field k and at its source lies the Kronecker--Weber Theorem, which solves this problem for k=Q, the field of rational numbers (L. Kronecker [,], H. Weber []). The first complete proof of it has been given by D. Hilbert [] (see also [], Satz 131). An exposition of the early proofs is given in O. Neumann []. An elementary proof can be found in M.J. Greenberg [] and a proof which uses local methods was given by I.R. Shafarevich []. This proof is exposed in [] and []. The problem of describing all Abelian extensions of an algebraic number field has been stated as the twelfth problem in the famous list of problems given by D. Hilbert [] in his lecture at the Second International Congress of Mathematicians in Paris in August 1900. The work of H. Weber [], D. Hilbert [], P. Furtwängler [], T. Takagi [] and E. Artin [] (subsumed in the report of H. Hasse []) in the first quarter of our century led to its solution. In it Abelian extensions of an algebraic number field k have been associated with certain ideal class-groups related to k. This classical approach will be described in Section 2.

The analogy between the theory of algebraic numbers and the theory of algebraic functions in one variable had already been observed in 1882 (R. Dedekind and H. Weber []). This led later to the theory of Dedekind domains and culminated in the theory of ideals in commutative rings. In 1931 F.K. Schmidt [] succeeded in constructing the analogue of the class-field theory for algebraic function fields in one variable over a finite field. His result gave a biunique correspondence between finite Abelian extensions

of a given algebraic function field k and certain subgroups of the divisor group of k.

Modern class-field theory begins with the invention of ideles by C. Chevalley [] who in C. Chevalley [] reinterpreted classical class-field theory in terms of ideles, using the theory of associative algebras. This approach led to a simultaneous proof of the class-field theory in both cases. (The book [] presents such a proof using algebras.) Later development eliminated the use of algebras in the proofs of the main results of class-field theory and replaced them by the formalism of cohomology. Expositions of this method are given in []. Recently a simplification in the theory was presented by J. Neukirch [,,], whose axiomatical approach reduced the whole problematics to purely group-theoretical reasonings utilizing only rudiments of cohomology. We shall sketch the main ideas in Section 3 and the next sections will be devoted to a choice of applications of the class-field theory.

Expositions of class-field theory can be found in [].

Generalizations of the class-field theory to the non-Abelian case (the Langlands program) and to other classes of fields are the subject of other chapters.