This article appeared in:

(Editors Tauno Metsänkylä and Paulo Ribenboim)

Queen's Papers in Pure and Applied Mathematics

Volume 91, 1992

Kingston Ontario Canada

For the mathematical community, Kustaa Inkeri is the author of significant papers on number theory, especially on topics related to Fermat's Last Theorem. Finnish mathematicians know Inkeri as the founder of the school of number theory in Finland. At the University of Turku, many of us still think of Inkeri as the Head of the Mathematics Department, a position he held for about 20 years.

The present contribution is intended to give a picture of the man behind these achievements. So this is an essay expressly about the person of a mathematician and no attempt will be made to describe or sum up Inkeri's mathematical work. For an appreciation of the latter the reader is asked to take advantage of the rich material in the rest of this volume.

Kustaa Adolf Inkeri was born at Laitila on November 12, 1908. Laitila is a parish (since 1986 a town), sixty kilometers to the north of Turku, in which the main source of livelihood is agriculture. Kustaa Inkeri's father was himself a farmer and had a family of five children. As an ambitious man Inkeri's father succeeded in bettering his family's position by moving every so often to new farms and for the Inkeri children these changes to new localities in different parts of south-western Finland meant that they grew up in a rather diverse set of environments.

Kustaa Inkeri took his school-leaving examination in Turku in 1928; the family had, in fact, settled near the city a few years earlier. After completing his compulsory period of military service, he enrolled in the Finnish University of Turku (nowadays the University of Turku). In the deepening economic depression of that time Inkeri had to work his way through college and he did this mainly by giving private lessons. While still a student, he married Fanny Ragnhild Vänni, the daughter of a businessman, and they had their first child, a son, even before the father (in 1936) had taken his master's degree. Four more children were later born to the marriage.

In addition to mathematics Inkeri took two other majors in his master's degree, namely in physics and astronomy. For a period he worked in the University Observatory. But having also undertaken a course of teacher-training, he obtained a post in a secondary school (at that time these posts were few and far between) in a country town not too distant from Turku.

The Second World War soon interrupted Inkeri's work as a school teacher. First came the Winter War of 1939--40 and then the Continuation War of 1941--44. Inkeri's service included a period as a company commander at the front, then he later became a staff officer. During a comparatively peaceful period of stationary warfare, the idea had arisen of organizing advanced teaching in various subjects for those interested, and Inkeri took part in this very unofficial activity by holding courses in mathematics corresponding to the first stage of university degree studies.

On returning to civilian life, Kustaa Inkeri took up a post in a Turku secondary school and at the same time worked as a part-time lecturer in the University of Turku. He was able to relinquish his secondary-school teaching upon taking his doctorate in 1946. Thereafter he soon became a docent and in 1950 was appointed to the chair of mathematics.

Inkeri recalls the enthusiasm with which he tackled mathematical problems in his early childhood. Hence, when he began his university studies, the choice of mathematics as his main subject was quite clear. Present generations of students would find it somewhat difficult to imagine the modest framework within which the university operated at that time. Throughout the whole university there were only about 450 students and in mathematics there was only one professor, Kalle Väisälä, and one other teacher. In this small world, however, the atmosphere for studying was all the more inspiring, and Väisälä was, moreover, an excellent teacher. His own special field was algebra, but number theory was a field that Väisälä took a close interest in, too. After following a special course on the theory of algebraic numbers and a further course particularly devoted to quadratic fields, Inkeri was hooked. For his master's thesis he wrote on the class number of the cyclotomic field and on Fermat's Last Theorem.

Inkeri became Väisälä's only doctoral student. Circumstances then were not very beguiling for those considering doctoral studies, for the chances of gaining a university post in mathematics were minimal. The number of posts of this kind, whether temporary or permanent, was slender in the extreme throughout the whole of Finland and the appointments-structure was an unhappy one. Thus in the University of Turku there was not a single assistant's post in mathematics. In 1947, however, an associate professorship was established and this was temporarily held by Inkeri until his appointment as full professor.

In the 1950s the university began to grow. New buildings were put up and for the first time the teaching staff in mathematics were given rooms of their own. This was the actual birth of the Department of Mathematics. Kustaa Inkeri was chairman of the Department until his retirement in 1972. The period of his leadership coincided with a dramatic rise both in the number of students and staff --- the great post-war age-groups were now coming to the university --- and by the end of the period the Department had almost attained its present size, with a personnel of about thirty, of whom three are full professors and four associate professors. For a time a part of the staff included the teachers of Applied Mathematics, but as this group grew in strength, it split off to form a department of its own.

In all these changing circumstances a great deal was demanded of a departmental head --- not merely his time but also his skill. Inkeri had undoubtedly done well, for when in the 1970s a passion to democratize the universities or, in other words, to transfer power to elected bodies took hold, the Department of Mathematics reacted quite coolly to the reform.

Although Professor Inkeri managed to avoid the burdensome post of, say, Dean of the Faculty, he was far from wishing to evade responsibility and was asked to hold many important commissions of trust. In these tasks he behaved with scrupulous objectivity, a quality which in learned circles is not always as evident as might be thought.

A well-liked departmental head must also be demanding and exact, which Inkeri certainly was. For example, it went without saying that in the Department of Mathematics all teaching had to be conscientiously undertaken. Inkeri prepared his own lectures thoroughly and they were a model of clarity. They began on the minute-stroke, but on the other hand it often turned out that the prescribed fifteen-minute break shrank to nothing and that the lectures also went on over the time-limit. In this manner was the proof of the theorem generally accomplished, but on those rare occasions when, let us say, time was found only to deal with three parts out of five of the proof, then in the subsequent lecture Inkeri would begin straightaway with the fourth part. Did someone once make a remark about professors being absent-minded?

The choice of topics dealt with in Inkeri's courses ranged from number theory and algebra to real and complex analysis. A correspondingly wide breadth of interest has been characteristic of his attitude towards research, whether in the doing of it himself or in following developments in the research field or in supervising the research of others. In addition to algebraic number theory Inkeri can be consulted on many aspects of analytic number theory and Diophantine approximations, among other matters. Of his former pupils, seven have taken their doctorates, six of these in the field of number theory. All seven now hold professorships in various Finnish universities. Inkeri himself --- modestly --- counts only four of these as his "real" doctoral students, the other three having found more independently subjects for their dissertations.

In his teaching Inkeri emphasized from his basic lecture-courses onwards the rigour of mathematical deduction. He also implanted among his students the need for healthy doubt: do not believe the proof unless you have yourself checked every step in it. In supervising doctoral work Inkeri did not force advice on the candidate, let alone try to goad or pressure. Yet in some indefinable fashion an awareness always developed in the candidate that this or that matter had to be down on paper by this or that time and when Inkeri got the paper in hand he would read it most carefully through, pointing out mistakes and making other comments, which would invariably turn out to be well-grounded. In no case did Inkeri either ever find it enough for the content of the work alone to be in shape; an impeccable clarity of presentation was also demanded.

It was natural for Kustaa Inkeri to be engaged not only in reviewing the work of his own doctoral students but also those of many another advanced student in Finland. Further, at the request of editors and of the writers themselves, he has read through numerous manuscripts for books and articles, among them, inevitably, a countless number of amateur attempts to solve Fermat's problem. While it is true that each one of us must, in greater or lesser degree, undertake review-work of this kind, in Professor Inkeri's case the admirable number of pages of text dealt with and, above all, the unusual care and exactitude with which he has approached these tasks, have constituted a commitment well worthy of note. Were we all to partake of the same standard of carefulness, printed mathematical texts would be perceptibly more readable than is nowadays the case!

In his university post Inkeri kept fit by going for long walks. Never once did he cancel his lectures for illness, nor indeed for any other reason, and he never took leave of absence from his post. His study trip to Göttingen in 1951 was arranged in the period May--August, which in the Finnish university system is the summer lecture-break. In parentheses it is worth mentioning that in Göttingen at that time Siegel, Deuring, and T. Schneider, inter alia, were active, and a warm welcome was given to the visiting Finnish mathematician.

From everything said above, the reader should not assume that Kustaa Inkeri is pedantically unapproachable and difficult to converse spontaneously with. Just the opposite is true. In private conversation Inkeri listens to his partner and shows a great understanding in respect of his or her problems. In any company, whether gathered together formally or informally, he is able, tactfully and sensitively, to give content and direction to the conversation and to relax any tension with a few well-chosen words.

As early as the year 1951 Kustaa Inkeri was elected to the Finnish
Academy of Science and Letters and he was later awarded one of the
Academy's prizes of honour. He has received several other distinctions
and marks of recognition. Thus for his sixtieth birthday a
*Festschrift* was published in his honour, which comprised
contributions written by his former pupils (Annales Universitatis
Turkuensis, Ser.A I 118, 1968). His portrait was also painted then.
How faithful a likeness the portrait is can be checked by any visitor
to the Department; in fact, the model who sat for the portrait may
readily be found nowadays, too, whether, for example, in the
departmental library or in the audience of the number-theory seminar.

Turku, July 13, 1992

**Tauno Metsänkylä**