Peter Roquette, Oberwolfach, March 2006 Peter Roquette Contributions to the History of Number Theory in the 20th Century Author: Peter Roquette Ruprecht-Karls-Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 69120 Heidelberg Germany E-mail: [email protected] 2010 Mathematics Subject Classification (primary; secondary): 01-02, 03-03, 11-03, 12-03 , 16-03, 20-03; 01A60, 01A70, 01A75, 11E04, 11E88, 11R18 11R37, 11U10 ISBN 978-3-03719-113-2 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. 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, re-use of illustrations, recitation, broad- casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2013 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 To my friend Günther Frei who introduced me to and kindled my interest in the history of number theory Preface This volume contains my articles on the history of number theory except those which are already included in my “Collected Papers”. All articles have been checked and reedited. Two articles which originally were written in German have been translated. I would like to thank all the people who have helped me to prepare this volume, foremost Keith Conrad and also Milena Hering who have streamlined my translation. Particular thanks go to my wife Erika who has meticulously read and corrected the whole manuscript. November 2012 Peter Roquette Contents Preface vii List of figures xi 1 The Brauer–Hasse–Noether Theorem 1 2 The remarkable career of Otto Grün 77 3 At Emmy Noether’s funeral 117 4 Emmy Noether and Hermann Weyl 129 5 Emmy Noether: The testimonials 163 6 Abraham Robinson and his infinitesimals 175 7 Cahit Arf and his invariant 189 8 Hasse–Arf–Langlands 223 9 Ernst Steinitz and abstract field theory 227 10 Heinrich-Wolfgang Leopoldt 239 11 On Hoechsmann’s Theorem 245 Acknowledgements 255 Bibliography 257 Name Index 273 Subject Index 277 List of Figures Richard Brauer, Helmut Hasse, Emmy Noether 2 Otto Grün 79 Noether’s tomb 125 Emmy Noether, Hermann Weyl 146 Emmy Noether 1933 165 Abraham Robinson 176 Cahit Arf 190 Helmut Hasse, Cahit Arf, Robert Langlands 225 Ernst Steinitz 227 Heinrich-Wolfgang Leopoldt 239 Chapter 1 The Brauer–Hasse–Noether Theorem Revised version of: The Brauer–Hasse–Noether theorem in historical perspective. Schriften der Math.-Phys. Klasse der HeidelbergerAkademie der Wissenschaften Nr. 15 (2005). 1.1 Introduction 1 1.2 The Main Theorem: cyclic algebras 3 1.3 The paper: dedication to Hensel 6 1.4 The Local-Global Principle 11 1.5 From the LGP to the Main Theorem 20 1.6 The Brauer group and class field theory 31 1.7 The team: Noether, Brauer and Hasse 45 1.8 The American connection: Albert 56 1.9 Epilogue: Käte Hey 73 1.1 Introduction The legacy of Helmut Hasse, consisting of letters, manuscripts and other papers, is kept at the Handschriftenabteilung of the University Library at Göttingen. Hasse had an extensive correspondence; he liked to exchange mathematical ideas, results and methods freely with his colleagues. There are more than 8 000 documents preserved. Although not all of them are of equal mathematical interest, searching through this treasure can help us to assess the development of Number Theory through the 1920s and 1930s. Unfortunately, most of the correspondence is preserved on one side only, i.e., the letters sent to Hasse are available whereas many of the letters which had been sent from him, often handwritten, seem to be lost. So we have to interpolate, as far as possible, from the replies to Hasse and from other contexts, in order to find out what he had written in his outgoing letters.1 1An exception is the correspondence between Hasse and Richard Brauer. Thanks to Prof. Fred Brauer, the letters from Hasse to Richard Brauer are now available in Göttingen too. 2 1 The Brauer–Hasse–Noether Theorem The present article is largely based on the letters and other documents which I have found concerning the Brauer–Hasse–Noether Theorem in the theory of algebras; this covers the years around 1931. Besides the documents from the Hasse and the Brauer legacy in Göttingen, I shall also use some letters from Emmy Noether to Richard Brauer which are preserved at the Bryn Mawr College Library (Pennsylvania, USA). We should be aware that the Brauer–Hasse–Noether Theorem, although to be rated as a highlight, does not constitute the summit and end point of a development. We have to regard it as a step, important but not final, in a development which leads to the view of class field theory as we see it today. By concentrating on the Brauer– Hasse–Noether Theorem we get only what may be called a snapshot within the great edifice of class field theory. A snapshot is not a panoramic view. Accordingly, the reader might miss several aspects which also could throw some light on the position of the Brauer–Hasse– Noether theorem, its sources and its consequences, not only within algebraic number theory but also in other mathematical disciplines. It would have been impossible to include all these into this paper. Thus I have decided to present it as it is now, being aware of its shortcomings with respect to the range of topics treated, as well as the time span taken into consideration. R. Brauer H. Hasse E. Noether A preliminary version of this article had been written in connection with my lecture at the conference March 22–24, 2001 in Stuttgart which was dedicated to the memory of Richard Brauer on the occasion of his 100th birthday. For Brauer, the cooperation with Noether and Hasse in this project constituted an unforgettable, exciting experience. Let us cite from a letter he wrote many years later, on March 3, 1961, to Helmut Hasse: … ist es 35 Jahre her, daß ich durch Sie mit der Klassenkörpertheorie bekannt geworden bin. Daß ich in Zusammenarbeit mit Ihnen und Emmy 1 The Brauer–Hasse–Noether Theorem 3 Noether ein wenig dazu beitragen konnte, ist auch mir eine der schönsten Erinnerungen, und ich werde die Aufregung der Tage, in denen die Arbeit entstand, nie vergessen. … it is now 35 years since you introduced me to class field theory. It belongs to my most delightful memories that I was able, in cooperation with you and Emmy Noether, to give some little contribution, and I shall never forget the excitement of those days when the paper took shape. The available documents indicate that a similar feeling of excitement was present also in the minds of the other actors in this play. Besides Hasse and Noether we have to mention Artin and also Albert in this connection. Other names will appear in due course. As toA.AdrianAlbert, he had an extended exchange of letters with Hasse, starting in 1931, on the Local-Global Principle for algebras. In the paper of Brauer–Hasse– Noether the authors explain that and how Albert had an independent share in the proof of the Main Theorem. Accordingly some people have suggested that perhaps it would be justified to include his name as an author, i.e., to talk about the “Albert– Brauer–Hasse–Noether Theorem”. But in this article we shall use the original name of the theorem, i.e., without Albert, since this has become standard in the literature. In Section 1.8 we will describe the role of Albert in the proof of the Brauer–Hasse– Noether theorem, based on the relevant part of the correspondence of Albert with Hasse. Acknowledgement. Preliminary versions had been on my homepage for some time. I would like to express my thanks to all who cared to send me their comments each of which I have carefully examined and taken into consideration. Moreover, I wish to thank Falko Lorenz and Keith Conrad for their careful reading, their corrections and valuable comments. Last but not least I would like to express my gratitude to Mrs. Nancy Albert, daughter of A.A.Albert, for letting me share her recollections of her father. This was particularly helpful to me while preparing Section 1.8. 1.2 The Main Theorem: cyclic algebras On December 29, 1931 Kurt Hensel, the mathematician who had discovered p- adic numbers, celebrated his 70th birthday. On this occasion a special volume of Crelle’s Journal was dedicated to him since he was the chief editor of Crelle’s Journal at that time, and had been for almost 30 years. The dedication volume contains the paper [BHN32], authored jointly by Richard Brauer, Helmut Hasse and Emmy Noether, with the title: Beweis eines Hauptsatzes in der Theorie der Algebren 4 1 The Brauer–Hasse–Noether Theorem Proof of a Main Theorem in the theory of algebras The paper starts with the following sentence: Endlich ist es unseren vereinten Bemühungen gelungen, die Richtigkeit des folgenden Satzes zu beweisen, der für die Strukturtheorie der Al- gebren über algebraischen Zahlkörpern sowie auch darüber hinaus von grundlegender Bedeutung ist: … At last our joint endeavours have finally been successful, to prove the following theorem which is of fundamental importance for the structure theory of algebras over number fields, and also beyond … The theorem in question, which has become known as the Brauer–Hasse–Noether Theorem, reads as follows: Hauptsatz.