Galois Theory of p-Extensions
First published in German in 1970 and translated into Russian in 1973, this classic now becomes available in English. After introducing the theory of pro-p groups and their cohomology, it discusses presentations of the Galois groups G S of maximal p-exten
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Springer-Verlag Berlin Heidelberg GmbH
Helmut Koch
Galois Theory of p-Extensions
Springer
Helmut Koch Humboldt-Universitat Berlin Institut fur Mathematik Unter den Linden 6 10117 Berlin Gemany
Translator Franz Lemmermeyer California State University San Marcos Department of Mathematics 333 South Twin Oaks Valley Rd. 92096-0001 San Marcos, CA USA e-mail: [email protected]
Catalog-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Koch, Helmut: Galois theory of p-extensions I Helmut Koch. ISBN 978-3-642-07817-0 ISBN 978-3-662-04967-9 (eBook) DOI 10.1007/978-3-662-04967-9
Mathematics Subject Classification (2000): 11R34, 11-02
This work is subject to copyright. Ali 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 microlilm 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-Verlag Berlin HeidelbergGmbH . Violations are liable for prosecution under the German Copyright Law. http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Originally published by Springer-Verlag Berlin Heidelberg New York in Softcover reprint of the hardcover lst edition 200.
2002
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. Typeset in TsX by the Translator Cover production: Erich Kirchner, Heidelberg SPIN 10868507
44/314.db - 5 4 3 2 1 o - Printed on acid-free paper
Foreword
This book deals with a quite young area of algebraic number theory: the algebraic theory of p-extensions, which was developed in the last 25 years and has now reached a degree of completion that makes a systematic presentation highly desirable. This area of arithmetic deals with the theory of finite extensions of fields of arithmetic type. These are p-adic number fields , fields of form·al power series with finite fields of constants, algebraic number fields and algebraic function fields of one variable with finite fields of constants. The main goal is obtaining information beyond what is provided by classical class field theory, which- as is well known - describes the extensions with commutative Galois group. The commutativity of the Galois group is very essential here. Class field theory is thereby closely connected with a wide circle of mathematical ideas ranging from the theory of radical extensions (nowadays called Kummer theory) to topological duality theorems, the theory of abelian and harmonic integrals and Picard varieties. The group-theoretical foundation of all these questions is the Pontryagin duality of abelian groups and their charact
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