The Theory of Classical Valuations
In his studies of cyclotomic fields, in view of establishing his monumental theorem about Fermat's last theorem, Kummer introduced "local" methods. They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi
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Paulo Ribenboim
The Theory of Classical Valuations
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Springer
Paulo Ribenboim Department of Mathematics and Statistics Queen's University Kingston, Ontario K7L 3N6 Canada
Mathematics Subject Classification (1991): 11-02, 11161, 13AI8, 16W60 Library of Congress Cataloging-in-Publication Data Ribenboim, Paulo. The theory of classical valuationsfPaulo Ribenboim p. cm. Includes bibliographical references and index. ISBN 978-1-4612-6814-7 ISBN 978-1-4612-0551-7 (eBook) DOI 10.1007/978-1-4612-0551-7 I. Aigebraic fields. 2. Valuation theory. I. Title. QA247.R483 1998 512'.3-dc21 98-4349 Printed on acid-free paper. © 1999 Springer Science+Business Media New York Originally published by Springer-Verlag New York Berlin Heidelberg in 1999 Softcover reprint of the hardcover 1st edition 1999
All rights reserved. This work rnay not be translated or copied in whole or in part without the written permission ofthe publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by sirnilar or dissirnilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, rnay accordingly be used freely byanyone. Production coordinated by Brian Howe and managed by Bill Imbomoni; manufacturing supervised by Joe Quatela. Typeset by Asco Typesetters, Hong Kong.
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ISBN 978-1-4612-6814-7
SPIN 10675904
Preface
This book contains a presentation of the theory of classical valuations. No details are spared (except in the Preface) so the text may be understood by graduate (and even undergraduate) students. It should also be profitable to nonspecialists wanting to become acquainted with the subject. Specialists may hopefully enjoy certain proofs. The study of valuations and absolute values was initiated by Kummer and Hensel, in connection with the arithmetic of number fields. Valuations provide a so-called local point of view and the family of all valuations allows a description of the ideals. The classical valuations are also attached to the places of algebraic curves. The treatment in this book is self-contained. The detailed exposition of the theory of decomposition, inertia, and ramification is not easily found (if at all) in any other Englishlanguage textbook. It is undoubtedly possible to teach the essential parts of this book in a one-semester course. The material may also be used in a graduate student's seminar or for independent study. It is our hope that the present volume will call the attention of mathematicians to the very beautiful and classical theory of valuations. August, 1998
Paulo Ribenboim
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