Universitätsverlag Göttingen Emil Artin und Helmut Hasse Die Korrespondenz 1923-1934 Herausgegeben und kommentiert von Günther Frei und Peter Roquette unter Mitwirkung von Franz Lemmermeyer Emil Artin and Helmut Hasse Their Correspondence 1923-1934 Edited and commented by Günther Frei and Peter Roquette with contributions of Franz Lemmermeyer With an Introduction in English Günther Frei und Peter Roquette Emil Artin und Helmut Hasse This work is licensed under the Creative Commons License 2.0 “by-nd”, allowing you to download, distribute and print the document in a few copies for private or educational use, given that the document stays unchanged and the creator is mentioned. You are not allowed to sell copies of the free version. erschienen im Universitätsverlag Göttingen 2008 Emil Artin und Helmut Hasse Die Korrespondenz 1923-1934 Herausgegeben und kommentiert von Günter Frei und Peter Roquette unter Mitwirkung von Franz Lemmermeyer Universitätsverlag Göttingen 2008 Bibliographische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliographie; detaillierte bibliographische Daten sind im Internet über <http://dnb.ddb.de> abrufbar. Kontakt/Contact Peter Roquette Universität Heidelberg roquette@uni-hd.de Dieses Buch ist auch als freie Onlineversion über die Homepage des Verlags sowie über den OPAC der Niedersächsischen Staats- und Universitätsbibliothek (http://www.sub.uni-goettingen.de) erreichbar und darf gelesen, heruntergeladen sowie als Privatkopie ausgedruckt werden. Es gelten die Lizenzbestimmungen der Onlineversion. Es ist nicht gestattet, Kopien oder gedruckte Fassungen der freien Onlineversion zu veräußern. Titelabbildung: Artin 1930er Jahre Foto N. Artin Titelabbildung: Hasse 1930er Jahre Aus: Abh. Math. Semin. Hamb. Univ. Bd. 10 Umschlaggestaltung: Margo Bargheer © 2008 Universitätsverlag Göttingen http://univerlag.uni-goettingen.de ISBN: 978-3-940344-50-2 Übersicht Teil I: Vorspann 1. Introduction 9 2. Emil Artin, his life and his work 24 3. Erinnerungen an Helmut Hasse 32 4. Reziprozität für Potenzreste 42 5. Klassenkörpertheorie 49 6. Zeittafel 53 Teil II: Die Briefe 1. Briefe Nr. 1-7: 1923–1926 59 2. Briefe Nr. 8-17: 1927 121 3. Briefe Nr.18-28: 1928–1929 251 4. Briefe Nr.29-49: 1930–1934 303 Teil III: Anhang 1. Namenverzeichnis 456 2. Stichwortverzeichnis 461 3. Literaturverzeichnis 471 4. Abbildungsverzeichnis 499 5 Teil I Vorspann 7 1 Introduction This book contains the full text of all letters from Emil Artin to Helmut Hasse, as they are preserved in the Handschriftenabteilung of the Göttingen University Library, written in the years 1923 – 1934. There are 49 such let- ters. Unfortunately, the corresponding letters from Hasse to Artin seem to be lost; Artin was known not to keep many of the letters and papers which he received. So we have supplemented Artin’s letters by detailed comments where we discuss their mathematical content, comparing this with a descrip- tion of the mathematical environment of Hasse and Artin, of the tendencies of the time, and of the relevant literature. In this way it will become possible for the reader to obtain some idea of the content of the corresponding letters from Hasse to Artin too. Artin and Hasse were among those who shaped modern algebraic number theory. They were of the same age, born in the year 1898. They belonged to the post-war generation of mathematicians who started their university education towards the end of and immediately after World War I, Artin in Leipzig with Herglotz (after a brief interlude in Vienna 1917 with Furtwäng- ler) and Hasse in Marburg with Hensel (after brief interludes in Kiel 1917 with Toeplitz, and in Göttingen 1918/19 with Hecke). They obtained their Ph.D. in the same year 1921, and their two dissertations were considered as ground breaking contributions to number theory. Artin’s thesis contained the theory of hyperelliptic function fields over a finite field of constants, and he formulated the analogue of the Riemann hypothesis for those fields which was later proved by Hasse in the elliptic case and by A. Weil in the case of function fields of arbitrary genus. Hasse’s thesis contained the Local-Global Principle for quadratic forms over the rationals which he later, in his habili- tation thesis, generalized to quadratic forms over arbitrary number fields. In Germany at that time, one had to pass one’s „ Habilitation “ in order to qualify for the position of professor at university. Both Artin and Hasse did their Habilitation in short succession, Hasse in 1922 in Marburg and Artin in 1923 in Hamburg. We have already mentioned above that Hasse’s Habilitation thesis contains the Local-Global Principle for quadratic forms over number fields. Artin’s Habilitation thesis is his seminal paper on his new L -series, which led him, among other results, to his Reciprocity Law. The mathematical careers of Artin and Hasse in the 1920s developed quickly and in remarkably parallel steps, from which one can conclude that they were considered by the mathematical community as leading scientists of equal, high standing. Let us explain: 9 10 In the fall of 1922 Hasse accepted the position of Privatdozent at the Uni- versity of Kiel which had been offered to him by Toeplitz. Actually, Toeplitz originally wanted to get Artin for this position but the latter was not able to accept; in a letter dated February 27, 1922 Artin wrote to Toeplitz that he had already accepted a stipend, for the summer semester of 1922, from Courant in Göttingen. 1 At that time Artin was in Göttingen as a post-doc. But a little later, in October 1922, Artin accepted a position in Hamburg offered to him by Blaschke. And in 1923, after his Habilitation , he became Privatdozent at Hamburg University. In 1923 there appeared the first joint paper of Artin and Hasse. In 1925 Hasse accepted an offer of a position as full professor at the University in Halle. This time again, Toeplitz had recommended Artin for this position but, for reasons unknown to us, Artin had not been taken into consideration by the nomination committee in Halle. 2 Shortly afterwards, in early 1926, the University in Münster had to fill a vacancy of full professor in Mathematics, and the proposal of the Faculty was, first Hasse and secondly Artin. Since Hasse had just moved to Halle, the position was then offered to Artin who, however, declined since Hamburg matched the offer and he was promoted to full professor in Hamburg. Thus now Artin and Hasse were for a time the youngest Mathematics professors in Germany. In 1928, the University of Breslau had to find a successor for the reti- ring Adolf Kneser (the father of Hellmut Kneser and grandfather of Martin Kneser). Like two years earlier in Münster, the Faculty in Breslau also pro- posed the names of both Artin and Hasse, but this time in the reverse order: first Artin and then Hasse. Neither of the two accepted; Artin remained in Hamburg and Hasse in Halle. In the same year Artin got another offer, this time from the University of Leipzig. Again he declined. At the same time A. Fraenkel, who was in Kiel at that time, tried to get Hasse back from Halle to Kiel. From the Fraenkel–Hasse correspondence one can see that Fraenkel had tried everything in his power to have the ministry of education extend an offer to Hasse for a position in Kiel. However the offer never came since the ministry was of the opinion that, instead, Hasse should be offered the position in Marburg after the retirement of Kurt Hensel, Hasse’s academic teacher. Indeed in 1930, Hasse moved from Halle to Marburg. In the same 1 This has been reported in the article [Rei07] by Karin Reich. In that article one can find more personal information about Artin. 2 Among the Hilbert papers kept in Göttingen we have found the draft of a letter of Hilbert to the Prussian ministry of education, written in March, 1925, where he recom- mended, among others, Hasse for the position in Halle but did not mention Artin. 1. INTRODUCTION 11 year, Artin obtained an offer from the ETH in Zürich as the successor of Hermann Weyl who had moved to Göttingen 3 . Again, Artin decided to stay in Hamburg. We have said above that Artin and Hasse were considered by the mathe- matical community as of equal standing, but this does not mean that they worked closely together, not even that their mathematical interests were identical. We see from their correspondence that they freely exchanged ma- thematical ideas and informed each other about recent results, mostly about class field theory and Reciprocity Laws which were the prominent topics of their discussion. But at the same time each of them also followed other lines of interest which are not mentioned in their letters, and each kept his own distinctive mathematical style. Let us point out that both Artin and Hasse belonged to what Yandell 4 has called the „Honors Class“, in the sense that each of them had solved one of the famous problems which Hilbert had presented in the year 1900 in his Paris lecture. Artin solved the 17th Hilbert problem which reads: . . . ob nicht jede definite Form als Quotient von Summen von For- menquadraten dargestellt werden kann. . . . whether every definite form may be expressed as a quotient of sums of squares of forms. Artin obtained the positive answer to this question through the theory of formally real fields which he had developed jointly with Otto Schreier. 5 Hasse had worked on the 11th Hilbert Problem: . . . Aufgabe, eine quadratische Gleichung beliebig vieler Variabeln mit algebraischen Zahlencoeffizienten in solchen ganzen oder gebro- chenen Zahlen zu lösen, die in dem durch die Coefficienten bestimm- ten algebraischen Rationalitätsbereiche gelegen sind. . . . to solve a given quadratic equation with algebraic numerical coefficients in any number of variables by integral or fractional numbers belonging to the algebraic realm of rationality determined by the coefficients. 3 See [FS92]. 4 See [Yan02]. 5 See [Art27b]. 12 Hasse obtained a criterion of solvability by means of his Local-Global-Principle which permitted the reduction of the question to the local case where it could be explicitly discussed, thanks to Hensel’s results about the p -adics. 6 There is another Hilbert problem whose solution has to be credited jointly to both Artin and Hasse, namely the 9th problem which concerns class field theory and reciprocity. The problem reads: Für einen beliebigen Zahlkörper soll das Reciprocitätsgesetz der ` -ten Potenzreste bewiesen werden, wenn ` eine ungerade Primzahl bedeu- tet und ferner, wenn ` eine Potenz von 2 oder eine Potenz einer un- geraden Primzahl ist. Die Aufstellung des Gesetzes, wie die wesentli- chen Hülfsmittel zum Beweise desselben werden sich, wie ich glaube, ergeben, wenn man die von mir entwickelte Theorie des Körpers der ` ten Einheitswurzeln 7 und meine Theorie 8 des relativ-quadratischen Körpers in gehöriger Weise verallgemeinert. For any number field the law of reciprocity is to be proved for the ` -th power residues, when ` denotes an odd prime, and further when ` is a power of 2 or a power of an odd prime. The law, as well as the means essential to its proof, will, I believe, result from suitably generalizing the theory of the field of ` -th roots of unity which I developed, and my theory of relatively quadratic fields. The first part, for an odd prime ` , had been solved by Furtwängler 9 and also by Takagi. 10 But for the case of higher prime powers there was no general approach in sight before Artin’s Reciprocity Law had opened the way. The implementation of Artin’s result for the deduction of explicit formulas for the Reciprocity Laws, in case of power residues for an arbitrary exponent, is due to Hasse; it is published finally in the second part of his Klassenkörperbericht . The Artin–Hasse correspondence documents that and how they cooperated to achieve this aim. The „ suitable generalization “ of Hilbert’s theory of relatively quadratic fields has turned out to be precisely Takagi’s class field theory, crowned with Artin’s Reciprocity Law. * * * * * 6 See [Has24a]. We also refer to [Fre01a] for a detailed description of how Hasse was led to the Local-Global Principle. 7 In Hilbert’s Zahlbericht [Hil97] Part 5. 8 In [Hil99] and [Hil02]. 9 See the literature mentioned in our Literaturverzeichnis. 10 In his second paper [Tak22]. 1. INTRODUCTION 13 We do not know when Artin and Hasse met for the first time. It may have been at the annual meeting of the German Mathematical Society (DMV) in September 1922 in Leipzig which both attended. Artin presented a talk on a problem from analysis and geometry which had arisen in a correspondence with his academic teacher Herglotz in Leipzig. Hasse did not give a talk, he just accompanied his academic teacher Hensel to the meeting. Artin and Hasse certainly met several times during the winter semester 1922/23. As said above, at that time Artin was in Hamburg and Hasse was in Kiel. The towns of Hamburg and Kiel in northern Germany are not too far apart from each other, about 100 km. The mathematicians in Kiel often went to the colloquium in Hamburg which was led by Blaschke and Hecke. On those occasions Hasse and Artin met and there developed a close mutual exchange of mathematical ideas. Artin learned from Hasse how the p -adic methods of Hensel could be successfully applied to number theoretical pro- blems, and Hasse was informed by Artin about the two great papers by Takagi on class field theory and on Reciprocity Laws. For both Artin and Hasse the encounter with Takagi’s papers turned out to be an important stimulus for their future work. They immediately realized the enormous potential of the main discovery of Takagi, namely that every abelian extension of a number field is a class field. In fact their whole correspondence, which centers around class field theory and reciprocity, can be regarded as reflecting their critical preoccupation with Takagi’s papers, and their attempt to further simplify, streamline and complete class field theory and to put it to work in number theory. In the Takagi biography by Honda [Hon76] the story is told how Artin and Hasse got hold of a copy of Takagi’s papers. Honda reports that Takagi had heard about a brilliant young mathematician in Göttingen with the name of Siegel. Subsequently Takagi sent to Siegel a reprint of his first great paper [Tak20] on class field theory. And Honda continues: One day, when Siegel was talking with Artin about number fields, he took out the reprint which Takagi had sent to him, and per- suaded Artin to read it. This was at the beginning of 1922. Artin borrowed the reprint from Siegel. He spent three weeks in reading it through. Later, in 1962, he told the present author [Honda] : I felt strong admiration for it. It was not difficult to understand, since it was written very clearly. Somewhat later in Hamburg, Artin showed Takagi’s papers to Hasse. This incident was recalled when Honda interviewed Hasse: 14 In 1923 Artin urged him [Hasse] to read the two papers of Takagi. Reading the first paper, Hasse was deeply fascinated by its genera- lity, its clearness, its effective methods, and its wonderful results. He was given an even stronger inspiration by the second paper. Takagi’s second paper [Tak22] deals with the Reciprocity Law for power residues of prime exponent. Thus, by 1923 there were three young mathematicians in Germany who had read and appreciated Takagi’s papers on class field theory: Siegel, Artin and Hasse. Each of them immediately started to integrate Takagi’s results into his work with striking results, and in this way these results became quickly known among mathematicians – although 3 years prior to this Ta- kagi had not met with any visible response when he presented his paper at the International Mathematical Congress in Straßburg where German ma- thematicians had not been admitted. As to Artin, already his 1923 paper on Galois L -series relied heavily on Takagi. Although his main idea, namely the construction of L -series for Galois extensions, was evidently inspired by the work of Hecke 11 , there was one important point where Artin had to use Takagi. This was when Artin tried to identify his new L -series in the abelian case with the classical L -series of Dirichlet and Weber 12 . In order to do this, he had to use Takagi’s class field theory which implied that for any abelian extension of number fields there exists an isomorphism of the corresponding ray class group of the base field with the Galois group. But this was not quite sufficient for Artin, for he needed the fact that there is a canonical isomorphism given by associating to every unramified prime of the base field its Frobenius automorphism in the Galois group. Artin could not yet prove this in 1923. Only in the special case of a cyclic extension of prime degree (and also of an extension composed of these) could he extract this from Takagi’s papers. But that was only temporary. Four years later in 1927, Artin could prove his general Reciprocity Theorem for arbitrary abelian extensions, thus putting his new L -series on a solid base and at the same time completing Takagi’s class field theory. This important result was hailed by Takagi as one of the most beautiful results of algebraic number theory 11 On the influence of Hecke on Artin’s work see [Fre01b]. 12 It was Heinrich Weber who had introduced and studied the L -series for congruence class groups in arbitrary number fields, in generalization of Dirichlet’s L -series in the case of Q . See [Fre89]. 1. INTRODUCTION 15 In fact, Artin’s Reciprocity Theorem completely changed our understanding of class field theory which today is seen as the key to much of algebraic number theory. In the Artin–Hasse letters we can observe the exciting story of its discovery, and how it was put to work immediately. As to Hasse, he realized that Takagi’s papers could be put to use in the study of explicit Reciprocity Laws which was his main interest in those days. Already on April 23, 1923 he reported to his academic teacher Hensel: . . . Außerdem habe ich gerade die Ausarbeitung eines Kollegs über die Klassenkörpertheorie von Takagi vor, die ich mit unseren Metho- den sehr schön einfach darstellen kann. . . . Furthermore, I am just preparing the manuscript for a course on Takagi’s class field theory, which I can develop quite nicely with our methods. Thus Hasse did what mathematicians often do, namely he gave a lecture course since he wished to learn more about the topic. When he mentions „our methods“ then he means the p -adic methods of Hensel which he, Hasse, was endeavoring to put into their proper place in algebraic number theory. At the time Hasse was mainly concerned with the theory of norm residues which Hensel had started to investigate with p -adic methods. Norm residues play an important role in class field theory, and so it seems to us that Hasse, in this letter to Hensel, meant that he can develop the theory of norm residues „quite nicely with our methods“. This is evident from Hasse’s papers from those years, among them the joint paper of Artin and Hasse which is discussed in the first 5 letters of their correspondence. (Later in the early 1930s, Hasse was indeed able to develop class field theory with essential use of p -adic methods but we have no indication whether already at the time of this letter, in 1923, he had definite ideas on how to achieve this.) Hasse delivered his lecture course on Takagi’s class field theory in Kiel in the summer semester 1924. The notes for this course were composed by Rein- hold Baer. 13 They provided the basis for Hasse’s famous report on Takagi’s class field theory which he had been asked to present at the DMV meeting in Danzig in September 1925, and whose Part I went into print in 1926 14 This article became known as „ Klassenkörperbericht “ and was regarded in 13 Later in 1928, Hasse brought Baer as an assistant professor to Halle. There developed a lifelong friendship between the two and their families, surviving the dark years of the Nazi period in Germany when Baer had to emigrate from Germany. 14 See [Has26a]. 16 line with Hilbert’s „ Zahlbericht “ of 1897. Hasse’s report was not meant to replace the Zahlbericht , as is sometimes claimed. Hasse’s aim was to amplify the latter by a survey of Takagi’s class field theory; he wished to give a useful guide for those who wanted to study the details, so as to avoid unnecessary detours. Let us cite a postcard from Bessel-Hagen to Hilbert, dated August 17, 1926, as an example of the reception of the Klassenkörperbericht : . . . In dem vor wenigen Tagen erschienenen Hefte des Jahresberichtes der D.M.V. befindet sich ein Bericht von Hasse über die Klassenkör- pertheorie, der so vorzüglich klar geschrieben ist und den ganzen Aufbau der Theorie mit einem nur die Hauptgedanken enthaltenden Skelett der Beweise so wundervoll herausschält, daß die Lektüre ein wahres Vergnügen ist und für das Eindringen in die Theorie jetzt wirklich alle Schwierigkeiten aus dem Wege geräumt sind . . . . . . A few days ago there appeared the latest issue of the Jahres- bericht D.M.V. which contains a report by Hasse on class field theory, and which is written in excellent clarity. The design of the whole theory is wonderfully uncovered by presenting the main ideas only while the proofs are reduced to their skeletons. Reading this article is a real pleasure; now all obstacles are eliminated which may have hampered access to the theory. . . The impact of Hasse’s report was remarkable. Since the proofs in the re- port were „ reduced to their skeletons “ (as Bessel-Hagen wrote), Hasse added an additional Part Ia which, responding to demand, contained full proofs. Now a whole generation of mathematicians started to learn class field theory through Hasse’s Klassenkörperbericht . Their names include Claude Cheval- ley, Jacques Herbrand, Max Deuring, Arnold Scholz, Olga Taussky, Shokichi Iyanaga, Max Zorn, perhaps Hermann Weyl, and many more, not to for- get Emmy Noether. Whereas formerly class field theory was the topic of a select few, Hasse’s report brought about its „popularization“ among mathe- maticians. This had the effect that during the next years the proofs of class field theory quickly became streamlined, simplified and shortened. Artin and Hasse took active part in this development; their correspondence provides ample witness for this. Within a decade, and finally with Chevalley’s idea to use ideles instead of ideals, class field theory got a new look which was considered more natural. Subsequent to Parts I and Ia of the Klassenkörperbericht , there followed Part II which contained the derivation of explicit Reciprocity Laws on the 1. INTRODUCTION 17 basis of Takagi’s class field theory. Hasse had almost completed Part II when he obtained the information about the successful proof of Artin’s Reciprocity Law. This caused Hasse to completely rewrite Part II where now he included Artin’s result and used it as a basis to derive all known Reciprocity Laws for power residues – in the spirit of Hilbert’s 11th problem for arbitrary exponents. The story of this is mirrored in the Artin – Hasse correspondence. * * * * * The progress of work in Mathematics 15 depends, as we all know, on the facilities for communication between mathematicians. Publication of papers is one way of communication but this is usually the final step only; most mathematicians prefer some amount of communication along the way, during work in progress. Hasse’s main medium of communication was letters. Hasse was an ardent letter writer. The Artin file is only one of many others in the Hasse legacy at the Handschriftenabteilung in Göttingen. Besides numerous manuscripts and notes there are more than 1600 letter files in the Hasse legacy. Of course, not all of them are of the same level of interest for the mathematician as is the Artin file, or the Noether file which we edited some time ago. But many are quite interesting, and in the future we intend to edit more from the Hasse legacy. Unfortunately Hasse’s own letters, when handwritten, are mostly lost whereas the majority of preserved letters are from his correspondence partners and addressed to him. But as in the case of Artin’s or Noether’s letters, by carefully reading the replies and considering the mathematical environment of Hasse at the time, one may be able to get a fair picture of how he worked, of his main ideas, aims and hopes, of his relation to colleagues and friends, and about his personality. His letters contain not only information about his results; he freely and openly talked about his ideas and the attempts to realize them, and conversely he asked for the opinions and advice of his correspondence partners. There was no hiding information in the back, and he never brought up questions of priority. We can observe all this in his correspondence with Artin too. Artin, on the other hand, was not as fond of letter writing; his main me- dium of communication was teaching and conversation: in groups, seminars and in smaller circles. We have many statements of people near to him des- cribing his unpretentious way of communicating with everybody, demanding quick grasp of the essentials but never tired of explaining the necessary. He 15 and not only in Mathematics 18 was open to all kind of suggestions, and distributed joyfully what he knew. He liked to teach, also to young students, and his excellent lectures, always well prepared but without written notes, were hailed for their clarity and beauty. When he wrote letters then most of the time it was in response to a letter received, and not always quickly. This we also observe here. Several times he apologizes to Hasse for his delay in answering, pretending other activities as the cause, but on the whole giving the impression that it took him some effort to take up the pen and write. In this situation it is quite interesting to read the letters between these two mathematicians which were quite different in temperament, different in mathematical style and different in their attitude towards letter writing. The fact that such an extensive correspondence had come about, sometimes with very high exchange frequency, is due to the fact that they had something to say to each other, upon a subject by which they were both fascinated. We may add that both shared similar ideas of mathematical beauty which they strove to realize in their work as much as possible. In our time there are numerous workshops, meetings, conferences, sympo- sia, colloquia etc. where people can exchange their knowledge and opinions, the year round all over the globe. And those who cannot attend may use e-mail. It is lucky that Artin and Hasse lived at a time when letter writing was still widely in use, and also that at least one of them, Hasse, had been able to save his correspondence files. This provides us with first hand know- ledge about mathematical developments which could have been gathered only partially from the published papers. * * * * * The exchange of letters between Artin and Hasse did not proceed evenly; sometimes the letters followed each other in short succession while sometimes there were years with no letters. The first 5 letters were written within a week, more precisely between July 7 and 12, 1923. Some time earlier Hasse had visited Hamburg and given a colloquium talk on his new results about the explicit Reciprocity Laws; this was on March 1, 1923. On that occasion Artin and Hasse got into a discussion about the so-called 2 nd supplement to the Law of Reciprocity, in case of a prime exponent ` , and they had agreed to continue this discussion when Artin would visit Hasse in Kiel. Artin’s letters were meant to prepare this visit which took place on the weekend July 14-16, 1923. As a result there