# Tadashi Nakayama

### Born: 26 July 1912 in Tokyo, Japan

Died: 5 June 1964 in Nagoya, Japan

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**Tadashi Nakayama**'s father was a leading scholar of Chinese classics. Tadashi was brought up in Tokyo where he first attended primary school and then entered the Musasi High School. His performance at this school was outstanding and, after graduating from the High School, he entered Tokyo Imperial University. There he was taught by Teiji Takagi who had done important work on algebraic number theory. However, perhaps even more of an influence on Nakayama than Takagi was his student Kenjiro Shoda who had studied in Germany with Issai Schur in Berlin and with Emmy Noether in Göttingen. In 1934 Nakayama had his first three papers published.

*Über das Produkt zweier Algebrenklassen mit zueinander primen Diskriminanten*was written jointly with Shoda while the other two,

*Ein Satz über p-adischer Schiefkörper*and

*Über die Definition der Shodaschen Diskriminante eines normalen einfachen hyperkomplexen Systems*were single-authored papers but still influenced by Shoda.

Nakayama graduated with his Rigakushi (diploma) from Tokyo Imperial University in 1935 and, in the same year, he was appointed as an assistant at Osaka Imperial University (now Osaka University). Shoda had been appointed as a professor in the Faculty of Science at Osaka University in 1933 so Nakayama's appointment there allowed them to easily continue their work together. In fact Nakayama published the book

*Local Class Field Theory*in the year he was appointed to Osaka. He was promoted to assistant professor at Osaka University in 1937 having by this time twelve papers in print including a second one written in collaboration with Shoda. Experience abroad was important to leading Japanese mathematicians in this period and Nakayama followed this pattern spending time in the United States having been invited to the Institute for Advance Study at Princeton.

Before Nakayama left for the United States he had been ill and had contracted tuberculosis. He had to have a medical examination before being allowed to go abroad but he persuaded the doctor who did the examination not to mention tuberculosis in his report. [After Nakayama's early death, many felt that this doctor was to blame for covering up his tuberculosis at a young age.] It was in September 1937 that Nakayama arrived in Princeton and there he met a number of leading algebraists such as Hermann Weyl, Emil Artin who had just emigrated to the United States, and Claude Chevalley who arrived in Princeton in 1938. Richard Brauer was by this time a professor at Toronto and he invited Nakayama to make two research visits to Toronto during his time in the United States, When he visited Brauer, he become inspired to work on group representations, publishing articles such as

*Some studies on regular representations, induced representations and modular representations*(1938) and

*A remark on representations of groups*(1938). While with Brauer, he also met Cecil J Nesbitt, Brauer's first doctoral student in Toronto, and Nakayama and Nesbitt collaborated on the paper

*Note on symmetric algebras*(1938).

In 1939 Nakayama published the first part of his paper

*On Frobeniusean Algebras*in the

*Annals of Mathematics*. It was in 1939, after spending two years in North America, that he returned to Osaka University. In 1941 he submitted the two parts of the paper

*On Frobeniusean Algebras*(the second part was published in the

*Annals of Mathematics*in 1941) for the degree of

*Rigakuhakushi*(doctorate) which was conferred on him in that year. In 1942 he left Osaka to take up an appointment as an associate professor at the Nagoya Imperial University which had been founded in 1939. He was promoted to a full professor after two years at Nagoya University [1]:-

His second book,During the difficult times of World War II he continued his pioneering works in mathematics.

*Lattice Theory*, was published during the war years in 1944. By this time he had nearly 50 publications to his name including his first with Goro Azumaya. Together with Azumaya, Nakayama worked on the representation theory of algebras, particularly Frobenius algebras. They wrote three joint papers on their work and published the book

*Algebra. Theory of rings*in Japanese in 1954. Masatoshi Gündüz Ikeda writes in a review:-

The first part of the book, written by Nakayama on the structure of rings, is similar to the famous text by Nathan Jacobson. Although the second part was written by Azumaya, it included much of their joint work on the representation theory of algebras.This book ... is devoted to a systematic presentation of the theory of associative rings, a subject which has developed remarkably since the1940's. The contents are divided into two parts. The first part, "general structural theory", written by the first author[Nakayama], consists of chapters1-12and is mainly concerned with the theory of rings without finiteness assumptions. The second part "algebras and representation theory", written by the second author[Azumaya], consists of chapters13-17and is mainly devoted to the theory of algebras and their representations.

Nakayama spent further time on extended research visits. In 1948 and 1949 he was in the United States, at the University of Illinois, and, during 1953 and 1955, he spent time at Hamburg University in Germany, and spent a second period at the Institute for Advanced Study at Princeton. He received many honours for his achievements. For example, in 1947 he received (with his collaborator Goro Azumaya) a Chubu Nippon Bunka Sho in recognition of his research in the theory of infinite dimensional algebras. It was for his research on the theory of rings and representations that the Japan Academy of Science awarded him a Gakushi-in shou (Japan Academy Prize) in 1953 and, ten years later, he was elected a member of the Japan Academy.

The obituary [1] lists six book and 122 papers written by Nakayama. This level of productivity is all the more remarkable when one realises that he achieved this despite severe health problems. As the tuberculosis became worse, he refused to give up his work and continued to come to the mathematics department at Nagoya University. He was so short of breath that chairs were placed on each floor so the he could rest while moving around the building. Even when he was eventually confined to bed he did not give up mathematics but read Grothendieck's work on algebraic geometry. He continued to publish research up to the time of his death, his last paper

*Class group of cohomologically trivial modules and cyclotomic ideals*appearing in the journal

*Acta Arithmetica*in 1964, the year in which he died from tuberculosis.

Given these heath problems and the remarkable research achievements that he managed to keep up, it is all the more amazing that he was able to devote time to other matters [1]:-

As to his character, we quote again from [1]:-In addition to his scientific and educational activities great importance must be attached to the indefatigable labors which he expended for the development of the Mathematical Institute of the Nagoya University. Then he was one of the founders and the editor-in-chief of the 'Nagoya Mathematical Journal'. He was also an always enthusiastic and active member of the Mathematical Society of Japan, under whose auspices he often lectured in many towns of this country. His widespread activity also included his work as the cooperating editor of the 'Proceedings' of the Mathematical Society of Japan and 'Acta Arithmetica', and as a reviewer for the 'Mathematical Reviews' and for the 'Zentralblatt für Mathematik'.

Finally we mention that his name is well-known today among algebraists for 'Nakayama's lemma' is named after him.He was a man of rare nobility of mind and great kindness whose life was consumed in incessant labors. His gracious spirit and his modest sympathetic and self-effacing personality will remain an unforgettable and shining example.

**Article by:** *J J O'Connor* and *E F Robertson*

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