Class Field Theory From Theory to Practice
Global class field theory is a major achievement of algebraic number theory, based on the functorial properties of the reciprocity map and the existence theorem. The author works out the consequences and the practical use of these results by giving detail
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Springer-Verlag Berlin Heidelberg GmbH
Georges Gras
Class Field Theory From Theory to Practice
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Georges Gras University of Franche-Comté Faculty of Sciences Laboratory of Mathematics and CNRS 16, route de Gray 25030 Besançon Cedex, France e-mail: [email protected] or [email protected] Translator of the original French manuscript Henri Cohen University of Bordeaux I Mathematics and Computer Sciences 351, Cours de la Libération 33405 Talence Cedex, France Library of Congress Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the DeutscheNationalbibliografie; detailed bibliographic data is available in the Internet at .
Mathematics Subject Classification (2000): 11RXX, 12R37, 11R29, 11R70, 11R34, 11S31, 11Y40
ISSN 1439-7382 ISBN 978-3-642-07908-5 ISBN 978-3-662-11323-3 (eBook) DOI 10.1007/978-3-662-11323-3 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law.
http://www.springer.de © Springer-Verlag Berlin Heidelberg 2003 Originally published by Springer-Verlag Berlin Heidelberg New York in 2003 Softcover reprint of the hardcover 1st edition 2003 The use of general descriptive names, registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg Typesetting: by the author Printed on acid-free paper SPIN 10883751
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Preface
This book is intended for students, for researchers, and for all those who are familiar with classical algebraic number theory. Its aim is to help in the practical use and understanding of the principles of global class field theory for number fields, without any attempt to give proofs of the foundations or their chronological appearance. It does give some historical landmarks, however, for a theory which began at the beginning of the twentieth century and involved many first-rank mathematicians. More precisely, we will assume the existence and functorial properties of local and global reciprocity maps, as well as the existence of the Galois correspondence of class field theory (between class groups of K and abelian extensions of K, called class fields). These results are essentially given in Theorems II.1.4 and II.3.3, as well as Theorems II.1.5 and II.3.5 of Chapter II
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