Quadratic Forms Over Semilocal Rings
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655 Ricardo Baeza
Quad ratic Forms Over Semilocal Rings
Springer-Verlag Berlin Heidelberg New York 1978
Author Ricardo Baeza Mathematisches Institut FB 9 Universitat des Saarlandes 0-6600 Saarbri.lcken
AMS Subject Classifications (1970): primary: lOC05, lOE04, lOE08 ISBN 3-540-08845-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08845-8 Springer-Verlag New York Heidelberg Berlin 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.
© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Preface
The algebraic theory of quadratic forms originates from the well-known paper [W] of Witt (1937), where he introduced for the first time the so called Witt ring of a field, i.e. the ring of all quadratic forms over a field with respect to a certain equivalence relation (compare §4, chap.I in this work). The study of this ring and related questions is essentially what we understand by the algebraic theory of quadratic forms. Thirty years after the appearance of Witt's work, Pfister succeeded in his important papers [Pf]1,2,3 in giving the first results on the structure of the Witt ring of a field of characteristic not 2. Since Pfister's work appeared twelve years ago, a lot researchs on this subject have been made. Lam succeeded in writing down many of these researchs in his fine book [L], which is perhaps today the best source to which a student of the algebraic theory of quadratic forms may turn for a comprehensive treatment of this subject. Besides of the theory over fields a corresponding theory of quadratic forms over more general domains has been growing up. We cite in particular Knebusch's work on the related subject of symmetric bilinear forms (see [K]1
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The present work deals with the algebraic theory of quadratic forms over semi local rings. We have tried to give a treatment which works for any characteristic, i.e. we do not make any assumption about 2. If 2 is not a unit, then in general quadratic forms behave better than bilinear forms, because the former have much more automorphisms (for example an anisotropic bilinear space over a field of characteristic 2 has only one automorphism). This fact has been exploited throughout in this work (see §3,S in chap. I I I and §3,4 in chap. IV) • Of course our results cannot go so far as in the field case, because over semi local rings we do not have to our disposal one of the most powerful methods of the theory over fields, namely the transcendental method. For example it would be very interesting to have an elementary proof (i.e. without transce
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