Travaux de Claude Chevalley sur la théorie du corps de classes: Introduction

Abstract

This article explains the contributions of Claude Chevalley to class field theory. His leading motivation on the subject seemed to be to give an "arithmetic proof" to the theory and to reveal the nature of the outstanding harmony of the Takagi-Artin class field theory, which had been established just at the time he started his research. His main achievements on the subject may have been the first arithmetic proof of the local class field theory without depending on the global theory, arithmetization of the global class field theory, and its generalization and presentation for infinite extensions by introducing ideles, which are now a kind of natural language in algebraic number theory. On the one hand, in this article we have attempted to provide rigorous mathematical description. On the other hand, although we have not demonstrated any proof, we have endeavored to show the development of the series of mathematical ideas that produced a variety of important concepts, bore fruit as class field theory, and then moved Chevalley to create his remarkable and influential works. © The Mathematical Society of Japan and Springer-Verlag 2006.