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Travaux De Claude Chevalley Sur La Théorie Du Corps De Classes: Introduction

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Travaux De Claude Chevalley Sur La Théorie Du Corps De Classes: Introduction Japan. J. Math. 1, 25–85 (2006) DOI: 10.1007/s11537-006-0502-5 Travaux de Claude Chevalley sur la theorie´ du corps de classes: Introduction Shokichi Iyanaga Received: 13 July 2005 / Accepted: 8 February 2006 Published online: 2 April 2006 c The Mathematical Society of Japan and Springer-Verlag 2006 Communicated by: Toshiyuki Kobayashi 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 presen- tation 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. Keywords and phrases: class field theory, idele, C. Chevalley Mathematics Subject Classification (2000): 11R37, 11-03, 01A61 Note by the Communicating Editor: In the summer of 1994, the author began writing this article, having been invited to contribute it as an introduction to the Collected Works of Claude Chevalley, Volume 1. The author finished the work by the spring of 2000 and was waiting for its publication, but for reasons unrelated to the article, publication of the Chevalley volume was cancelled in 2005. Informed of the cancellation, Professor Iyanaga then chose to submit the article to this journal, the Japanese Journal of Mathematics, on July 13, 2005. (Toshiyuki Kobayashi on behalf of the author, who has been in hospital since the autumn of 2005.) S. IYANAGA Emeritus Professor, University of Tokyo, and member of the Japan Academy of Sciences.
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