Introduction to Quadratic Forms
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Herausgegeben von J. L. Doob A. Grothendieck E. Heinz F. Hirzebruch E. Hopf W. Maak S. MacLane W. Magnus J. K. Moser M. M. Postnikov F. K. Schmidt D. S. Scott K. Stein GeschäftsfUhrende Herausgeber B. Eckmann und B. L. van der Waerden
0. T. O'Meara
Introduction to Quadratic Forms
Third Corrected Printing
With 10 Figures
Springer-Verlag Berlin Heidelberg GmbH 1973
AMS Subject Classifications (1970) Primary 1002, 10 B 40, 10 C 05, 10 C 20. 10 C 30, 10 E 45. 1202, 12A10, 12A40, 12A45, 12A50, 12A90, 13C10, 13F05, 13F!O, !502, !5A33, !5A36, 15A57. 15A63, 15 A 66, 20 G 15, 20 G 25, 20 G 30, 20 G 40, 20 H 20, 20 H 25, 20 H 30, Secondary 12 A 65, 12 Jxx
by Springer-Verlag Berlin Heidelberg 1963, 1971, 1973 Originally published by Springer-Verlag Berlin Heidelberg New York in 1973
(C)
Softcoverreprint of the hardcover 3rd edition 1973 ISBN 978-3-662-41922-9 (eBook) ISBN 978-3-662-41775-1 DOI 10.1007/978-3-662-41922-9
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. re-use of illustrations, broadcasting. reproduction by photocopying machine or similar means, and storage in data banks. Under §54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. Library of Congress Catalog Card Number 73-10 50.1
In Memory of my Parents
Preface The main purpose of this book is to give an account of the fractional and integral dassification problern in the theory of quadratic forms over the local and global fields of algebraic nurober theory. The first book to investigate this subject in this generality and in the modern setting of geometric algebra is the highly original work Quadratische Formen und orthogonale Gruppen (Berlin, 1952) by M. EICHLER. The subject has made rapid strides since the appearance of this work ten years ago and during this time new concepts have been introduced, new techniques have been developed, new theorems have been proved, and new and simpler proofs have been found. There is therefore a need for a systematic account of the theory that incorporates the developments of the last decade. The dassification of quadratic forms depends very strongly on the nature of the underlying domain of coefficients. The domains that are really of interest are the domains of nurober theory: algebraic nurober fields, algebraic function fields in one variable over finite constant fields, all completions thereof, and rings of integers contained therein. PartOne introduces these domains via valuation theory. The nurober theoretic and function theoretic cases are handled in a unified way using the Product Formula, and the theory is developed up to the Dirichlet Unit Theorem and the finiteness of dass number. It is hoped that this will be of service, not only to the reader who is interested in quadratic forms, but also to the reader who wishes to go deeper into algebraic nurober theor
