Valued Fields
Absolute values and their completions -like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible
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Antonio J. Engler Alexander Prestel
Valued Fields
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Antonio J. Engler
Alexander Prestel
Departamento de Matemática IMECC-UNICAMP Cx. Postal 6065 13083-970 Campinas Brazil E-mail: [email protected]
Fak. Mathematik Fachbereich Mathematik und Statistik Postfach 5560 78457 Konstanz Germany E-mail: [email protected]
Library of Congress Control Number: 2005930440 Mathematics Subject Classification (2000): 12J10, 12J20 ISSN 1439-7382 ISBN-10 3-540-24221-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-24221-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, 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. Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
SPIN: 10996843
41/TechBooks
543210
To the memory of Otto Endler (17.9.1929–12.5.1988)
Preface
The purpose of this book is to give a self-contained and comprehensive introduction to the theory of general valuations, in contrast to classical absolute values. In particular, we present some applications of the general theory going beyond the use of absolute values. The book does not aim for an encyclopaedic presentation, but rather prefers a streamlined style, leading eventually to deep results of recent research. While the classical theory of absolute values can be found in many books, in particular those on number theory, there are few textbooks devoted to the general theory of valuations. To our knowledge, these are O. Schilling (1950, [27]), P. Ribenboim (1965, [23]), and O. Endler (1972, [6]). Besides those, one can find, however, chapters on general valuation theory in several books, such as in O. Zariski – P. Samuel (1960, [33]) or Y. Ershov (2001, [8]). Concerning the history of valuation theory, the reader is referred to P. Roquette [25]. Both authors of this book have been deeply influenced by the late Otto Endler – the first author as a student, the second as a colleague. It was at the IMPA in Rio de Janeiro where we all met in the mid seventies. Since then we became followers of Krull’s development of henselia
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