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 letters. 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 description 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 Furtwängler) 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 habilitation 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:

In the fall of 1922 Hasse accepted the position of Privatdozent at the University 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 retiring Adolf Kneser (the father of Hellmut Kneser and grandfather of Martin Kneser). Like two years earlier in Münster, the Faculty in Breslau also proposed 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 year, Artin obtained an offer from the ETH in Zürich as the successor of Hermann Weyl who had moved to Göttingen³ . Again, Artin decided to stay in Hamburg.

We have said above that Artin and Hasse were considered by the mathematical 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 mathematical 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 Yandell⁴ 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:

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:

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:

The first part, for an odd prime , had been solved by Furtwängler⁹ 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.

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 problems, 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:

Somewhat later in Hamburg, Artin showed Takagi’s papers to Hasse. This incident was recalled when Honda interviewed Hasse:

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 Takagi had not met with any visible response when he presented his paper at the International Mathematical Congress in Straßburg where German mathematicians 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 Hecke¹¹ , 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 Weber¹² . 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

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:

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 Reinhold 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 1926¹⁴ . This article became known as „Klassenkörperbericht“ and was regarded in 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:

The impact of Hasse’s report was remarkable. Since the proofs in the report 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 Chevalley, Jacques Herbrand, Max Deuring, Arnold Scholz, Olga Taussky, Shokichi Iyanaga, Max Zorn, perhaps Hermann Weyl, and many more, not to forget Emmy Noether. Whereas formerly class field theory was the topic of a select few, Hasse’s report brought about its „popularization“ among mathematicians. 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 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 9th problem for arbitrary exponents. The story of this is mirrored in the Artin – Hasse correspondence.

The progress of work in Mathematics¹⁵ 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 medium of communication was teaching and conversation: in groups, seminars and in smaller circles. We have many statements of people near to him describing his unpretentious way of communicating with everybody, demanding quick grasp of the essentials but never tired of explaining the necessary. He 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, symposia, 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 knowledge 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 emerged their first joint paper, which appeared 1925 in Crelle’s Journal.¹⁶

There followed a period of 2 years without any letter exchanged. This does not mean that there was no communication between Artin and Hasse. On the contrary: Hasse visited frequently the Hamburg seminar and so there were many occasions for discussion and exchange of ideas, on reciprocity and other topics. In Hasse’s mathematical diary he refers several times to discussions with Artin.

This period came to an end when Hasse got a professorship in Halle, which was in the summer semester 1925. In the fall of that year Artin wrote again, telling that he had returned to number theory from what he called „going astray on topological roads“ (auf topologischen Abwegen). Apparently Artin wished to take up again his discussion with Hasse on reciprocity, looking for ideas towards a proof of his general Reciprocity Law which, as we have mentioned above already, he had formulated already in 1923 in his L-series paper.

The breakthrough came in 1927. Artin proudly informed Hasse on July 17, that recently in a lecture course he had given in Hamburg, his efforts had been successful and he had obtained a proof. Hasse was excited and replied immediately, and next day already Artin sent him the full proof. There followed a period of intense mutual correspondence. Within three weeks Artin wrote 7 letters to Hasse and we may safely assume that there were at least as many letters from Hasse to Artin. They discussed the implications of Artin’s Reciprocity Law, in particular the principal ideal theorem, the class field tower problem and the possible generalization of class field theory from the abelian case to arbitrary Galois extensions. Hasse was particularly interested in how to use Artin’s Reciprocity Law for the derivation of explicit Reciprocity Laws for power residues. Now, thanks to Artin’s result, they could be obtained for an arbitrary exponent, just as Hilbert had envisaged in his Paris lecture.

Their correspondence continued at a little slower pace, but still on a high mathematical level, until November 1932. For the reader of these letters there unfolds the exciting story of the emergence of new insights into class field theory, including the Local-Global-Principle for simple algebras and its use for a new proof of Artin’s Reciprocity Law. We see how Hasse, following a question of Artin, discovered local class field theory, first as a consequence of Artin’s global Reciprocity Law but later, on the suggestion of Emmy Noether, on a purely local basis. Their attempts to generalize class field theory to the case of non-abelian Galois extensions failed but this led to the beginning of algebraic cohomology which later culminated in the Artin-Tate version of class field theory.

In addition, we see the emergence of Part II of Hasse’s Klassenkörperbericht developing all known reciprocity formulas and much more, based on Artin’s Reciprocity Law. This book constitutes a monumental work marking the completion of an era having its roots deep in the past, back to the early 19th century. When Artin got the proof sheets he replied to Hasse that he had read it „with great pleasure“ (mit großem Vergnügen) – and he was promptly inspired to look for explicit formulas for the local contributions of his L-series at the ramified primes; this he had avoided in his first L-series paper and it did not appear in Hasse’s book. From this emerged Artin’s famous paper on conductors and the structure of the discriminant of Galois extensions – in close contact with Hasse whose contribution led to what was later called the Hasse-Arf theorem.

Hecke wrote later about the Klassenkörperbericht, in a letter to Hasse dated November 16, 1938:

In November 1932, Hasse gave a colloquium talk in Kiel on the number of solutions of binary diophantine congruences, in generalization of results of Davenport and Mordell. Immediately after this he followed an invitation by Artin to Hamburg and gave a talk at Artin’s seminar on the same topic. At this occasion Artin reminded him¹⁷ that the problem was essentially equivalent to the Riemann hypothesis for hyperelliptic function fields over finite fields which Artin had stated in his Ph.D. thesis. Thereafter it took Hasse less than three months to arrive at the proof in the case of elliptic fields. This happened at the end of February 1933.¹⁸ We are inclined to believe that Hasse had informed Artin about this result since, after all, it had been Artin who had put Hasse on the track with the zeta function for function fields. However, we have found no letter from Artin to Hasse in the year 1933.

But one year later, on January 17, 1934 Artin invited Hasse again to Hamburg and wrote:

Of course, Artin did not mean the classical Riemann hypothesis but its analogue for function fields over finite base fields. In a follow-up letter a few days later Artin wrote:

Hasse accepted Artin’s invitation and gave a compact lecture course of one week in Hamburg, consisting of 4 two-hour lectures, between February 5 and 9, 1934. He reported about his new method of constructing the endomorphism ring of an elliptic function field in characteristic p, and how to prove the Riemann hypothesis by studying the Frobenius operator. Thereafter Hasse wrote to Davenport:

But it turned out that it took more than patience to arrive at a proof of the Riemann hypothesis for function fields of arbitrary genus.

We see that Artin and Hasse at that time still vividly discussed mathematical problems of common interest. However, there were no more letters exchanged. We do not know the reason for the ensuing silence between the two. One explanation which offers itself is the deterioration of the political situation in Germany which had consequences also in academic life. Artin seemed to be worried because his wife Natascha was of Jewish origin (her father was Jewish) and she had to suffer the harassments of the time. Hasse was deeply concerned about the enforced exodus of so many mathematicians. He considered this as a fatal loss for Germany and did what he could do to act against this, but without much success. When in 1937 Hasse heard about plans of Artin to emigrate, he tried, on the one hand to obtain better and safe working conditions for Artin from the ministry of education, and on the other hand to persuade Artin to stay in Germany. He could not succeed under the circumstances and the Artins left Germany in 1937.¹⁹

As said above, we have supplemented Artin’s letters by detailed comments. We have given these comments in the language of Artin and Hasse, i.e., in German. Short comments are added in the form of footnotes to the respective letter. More detailed comments are to be found immediately after the letter with specific headings, which also show up in the table of contents.

Most of the letters deal with class field theory and with Reciprocity Laws for power residues. In order to understand the letters it is necessary to recall the state of the art at the time, and the terminology which was used by Artin and Hasse. Therefore we have included in the Vorspann two explanatory sections, one for the Reciprocity Laws of power residues (section 4) and the other for class field theory (section 5).

Also, we have included in the Vorspann two documents which contain personal recollections to Artin and to Hasse respectively. For Artin, we have chosen the obituary written by Hans Zassenhaus, Artin’s doctoral student in Hamburg (section 2). For Hasse, there is in section 3 a report about the early work on the present edition, and this contains personal recollections to Hasse. We believe that those two sections may help the reader to obtain a lively picture of the personalities of Artin and of Hasse whose mathematical ideas he will find in their letters.

The Zeittafel in section 6 may be useful to the reader, as an overview of the development of ideas and results which are discussed in the letters.

ACKNOWLEDGEMENT: As reported in section 3, the present book is the outcome of many years’ work (with interruptions), based on the first preliminary version back in 1981 [Fre81a]. We would like to thank all people who have given us their advice and encouragement during the writing of the book, and who have helped us with critical comments. Above all we would like to mention Franz Lemmermeyer who has followed the preliminary versions of the book and has lent us his invaluable advice and detailed historical knowledge in order to straighten up many important points. Patrick Morton has read the introduction and been of help to streamline our English. Last but not least we are glad that the Deutsche Forschungsgemeinschaft and the Möllgaard-Stiftung have provided financial support.

We hope that the reader will enjoy the book as we have enjoyed our work. Nevertheless, as in any work of this kind and size, there may have occurred errors, misprints, omissions or other shortcomings. Any comment or correction will be welcomed.