Quadratic and Hermitian Forms over Rings
From its birth (in Babylon?) till 1936 the theory of quadratic forms dealt almost exclusively with forms over the real field, the complex field or the ring of integers. Only as late as 1937 were the foundations of a theory over an arbitrary field laid. Th
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Editors
M. Artin S. S. Chern 1. Coates 1. M. Frohlich H. Hironaka F. Hirzebruch 1. Hormander S. MacLane C. C. Moore 1. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai 1. Tits M. Waldschmidt S.Watanabe Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
Max-Albert Knus
Quadratic and Hermitian Forms over Rings
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona
Max-Albert Knus Department Mathematik ETH-Zentrum CH-8092 Zurich Switzerland
Mathematics Subject Classification (1980): lOC05, 13-XX, 14F05, 14F15, 16-XX, 18F25, 18G50, 20GXX
ISBN-13: 978-3-642-75403-6 e-ISBN-13: 978-3-642-75401-2 DOl: 10.1007/978-3-642-75401-2 Library of Congress Cataloging in Publication Data. Knus, Max-Albert. Quadratic and hermitian forms over rings/Max-Albert Knus. p. cm. - (Grundlehren der mathematischen Wissenschaften; 294) Includes bibliograohical references and index. ISBN-13: 978-3-642-75403-6 1. Forms, Quadratic. 2. Hermitian forms. 3. Commutative rings. I. Title. II. Series. QA243.K55 1991 512'.74 - dc20 90-42429 CIP This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Softcover reprint of the hardcover 1st edition 1991 Typesetting: Macmillan India, Ltd., Bangalore, India 4113140-543210 Printed on acid-free paper
Foreword
From its birth (in Babylon?) till 1936 the theory of quadratic forms dealt almost exclusively with forms over the real field, the complex field or the ring of integers. Only as late as 1937 were the foundations of a theory over an arbitrary field laid. This was in a famous paper by Ernst Witt. Still too early, apparently, because it took another 25 years for the ideas of Witt to be pursued, notably by Albrecht Pfister, and expanded into a full branch of algebra. Around 1960 the development of algebraic topology and algebraic K-theory led to the study of quadratic forms over commutative rings and hermitian forms over rings with involutions. Not surprisingly, in this more general setting, algebraic K-theory plays the role that linear algebra plays in the case of fields. This book exposes the theory of quadratic and hermitian forms over rings in a very general setting. It avoids, as far as possible, any restriction on the characteristic and takes full advantage of the functorial aspects of the theory. The advantage of doing so is not only aesthetical: on the one hand, some classical proofs gain in simplicity and transparency, the most notable examples being the results on low-dimensional spinor groups; on the other hand new resu
