The Brauer-Hasse-Noether Theorem in Historical Perspective

The unpublished writings of Helmut Hasse, consisting of letters, manuscripts and other papers, are kept at the Handschriftenabteilung of the University Library at Göttingen. Hasse had an extensive correspondence; he liked to exchange mathematical ideas, r

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Peter Roquette

The Brauer-Hasse-Noether Theorem in Historical Perspective

Prof. Dr. Dr. h.c. Peter Roquette Mathematisches Institut der Universität Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg, Germany [email protected]

Library of Congress Control Number: 2004111361 ISBN 3-540-23005-X 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 use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany Cover design: Erich Kirchner, Heidelberg Typeset in LATEX by the author and edited by PublicationService Gisela Koch, Wiesenbach, using a modified Springer LATEX macro-package. Printed on acid-free paper 08/3150 hs – 5 4 3 2 1 0

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 The Main Theorem: Cyclic Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3 The Paper: Dedication to Hensel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4 The Local-Global Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.1 The Norm Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 The Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Factor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 From the Local-Global Principle to the Main Theorem . . . . . . . . . . . . . 25 5.1 5.2 5.3 5.4

The Splitting Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Unproven Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Grunwald-Wang Story . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The Weak Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Algebras with Pure Maximal Subfields . . . . . . . . . . . . . . . . . . 5.4.4 Exponent = Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Grunwald-Wang in the Setting of Valuation Theory . . . . . . . .

25 27 29 31 31 31 33 34 35

6 The Brauer Group and Class Field Theory . . . . . . . . . . . . . . . . . . . . . .