Value Functions on Simple Algebras, and Associated Graded Rings
This monograph is the first book-length treatment of valuation theory on finite-dimensional division algebras, a subject of active and substantial research over the last forty years. Its development was spurred in the last decades of the twentieth century
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Jean-Pierre Tignol Adrian R. Wadsworth
Value Functions on Simple Algebras, and Associated Graded Rings
Springer Monographs in Mathematics
More information about series at www.springer.com/series/3733
Jean-Pierre Tignol r Adrian R. Wadsworth
Value Functions on Simple Algebras, and Associated Graded Rings
Jean-Pierre Tignol ICTEAM Institute Université Catholique de Louvain Louvain-la-Neuve Belgium
ISSN 1439-7382 Springer Monographs in Mathematics ISBN 978-3-319-16359-8 DOI 10.1007/978-3-319-16360-4
Adrian R. Wadsworth Department of Mathematics University of California, San Diego La Jolla, California USA
ISSN 2196-9922 (electronic) ISBN 978-3-319-16360-4 (eBook)
Library of Congress Control Number: 2015937371 Mathematics Subject Classification: 16W60, 16W50, 16K20, 16K50, 16-03 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
The theory of finite-dimensional division algebras witnessed several breakthroughs in the latter decades of the twentieth century. Important advances, such as Amitsur’s construction of noncrossed product division algebras and Platonov’s solution of the Tannaka–Artin problem, relied on an inventive use of valuation theory, applied in the context of noncommutative rings. The subsequent development of valuation theory for finite-dimensional division algebras led to significant simplifications of the initial results and to a host of new constructions of division algebras satisfying various conditions, which shed much light on the structure of these algebras. In this research area, valuation theory has become a standard tool, for which this book is intended to provide a useful reference. The theory of valuations and valuation rings has been extended to division rings in several different ways. We treat here only the most stringent of these exten
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