In 1891 Takagi began the third stage of his schooling which he took at the Third High School in Kyoto. There were, at that time, eight academies and the brightest pupils went to the one corresponding to the area in which they lived in order to prepare for a university education. Takagi therefore, after showing great talents at middle school, made the natural progression to Kyoto where he studied for three years. In 1894 he graduated for the Third High School and entered Tokyo University, the only university in Japan at that time.
At Tokyo University Takagi took courses on calculus and analytic geometry. However he learnt more advanced mathematics by reading books rather than from lecture courses which he attended. He learnt about algebraic curves from George Salmon's book and he also studied Serret's Algèbre Supérieure Ⓣ. He eagerly read Heinrich Weber's Algebra text when it arrived in Japan and by 1898 Takagi had published his first paper. The paper shows a remarkably modern approach to algebra, very surprising for someone who had learnt most of his mathematics from textbooks. The paper begins:-
In looking back at the history of branches of mathematics, we see that they start with special and concrete beginnings and proceed by generalisation as they advance. This is manifested, for instance, in the theory of groups, which is one of the most important fields of present day mathematics, and is related to various other branches. It started as the theory of permutation groups, but now the general theory of groups does not suppose that elements of groups should be permutations. As Cayley has remarked, one has only to suppose that composition of elements satisfies certain laws ... We ... hope that the reader has understood that the essential point in algebra does not lie in the nature of the elements (which are not necessarily numbers) but in the way elements are composed.The paper also in noteworthy for containing an abstract definition of a field.
Takagi graduated from Tokyo University in 1897, and in the following year he was chosen as one of twelve students from Japan to study abroad. He sailed to Germany where he studied courses given by Fuchs, Frobenius and Schwarz at Berlin University. However, to his surprise, he discovered that he already knew most of the mathematics in these courses from the books that he had read back in Japan. He then read Hilbert's Zahlbericht Ⓣ, a report on algebraic number theory which had been published in 1897. Takagi wrote to Hilbert who arranged accommodation for him in Göttingen in a house in which he himself had previously lived. In  Takagi wrote:-
At the time when I studied in Germany, Göttingen was perhaps the only place in the world where research in algebraic number theory was going on. Thus, when I told Hilbert that I wanted to study this theory, he did not seem to believe me immediately ...If Takagi expected Hilbert to be actively engaged in algebraic number theory then he would have been disappointed. Hilbert had left this topic immediately after writing the Zahlbericht and by the time Takagi reached Göttingen he was engaged in studying the foundations of geometry and then integral equations. Although Hilbert was not directly involved with Takagi's research, the topic he worked on was certainly one that Hilbert considered of the utmost importance for it was a special case of what became Hilbert's 12th problem in his Paris lecture of 1900. In 1901 Takagi left Göttingen and returned to Japan where he was appointed as Assistant Professor in Algebra in the Department of Mathematics at Tokyo University. He married Toshi Tani in 1902 and they had three sons and five daughters. He completed a doctorate in Tokyo in 1903 presenting a thesis based on work he had undertaken in Göttingen. He was promoted to full professor in Tokyo University in 1904; he held this post until he retired in 1936.
On his return to Tokyo in 1903 Takagi proved a conjecture on abelian extensions of imaginary number fields made by Kronecker. Kronecker described this conjecture as:-
... the dearest dream of youth.Although Takagi was enthusiastic about research he did not continue to develop the work that he had begun in his thesis. He began to write textbooks, which of course were important for the development of Japanese mathematics both at school and university level. The first of these texts was A new course of arithmetic published in 1904. The 500 page work developed real numbers via Dedekind cuts. it was the first of many texts that Takagi wrote: between 1904 and 1911 he wrote 13 texts, but many were multi-volume works so the total number of volumes amounted to 20.
In  Takagi describes how he was led to begin research again:-
I am by nature someone who needs stimulus in order to work. there are now quite a number of Japanese mathematicians, but in those days, we had few colleagues. I did not have a heavy workload so you might imagine that I did research on class field theory in those carefree days, but it was not so.Takagi then went on in  to describe the ideas that relaunched his research career:-
The First World war started in 1914 and this gave me a stimulus, rather a negative stimulus. No scientific information reached Europe for four years. some said this would be the end of Japanese science while newspaper articles wrote of their sympathy for Japanese professors losing their jobs. This made me realise the obvious truth that every researcher had to be independent. Possibly I would have done no research for myself had it not been for World War I.
Concerning class field theory, I confess that I was misled by Hilbert. Hilbert considered only unramified class fields. from the standpoint of the theory of algebraic functions which are defined by Riemann surfaces, it is natural to limit consideration to the unramified case ... after the end of scientific exchange between Japan and Europe ... I was freed from that idea and suspected that every abelian extension might be a class field if the latter is not limited to the unramified case. I thought at first that this could not be true. were it false the idea should contain an error and I tried my best to find this error. At that time I almost suffered from a nervous breakdown. I dreamt often that I had resolved the question. I woke up and tried to remember my reasoning but in vain. I tried my utmost to find a counterexample to the conjecture which seemed all too perfect. finally I made my theory confirming this conjecture, but I could not rid myself of the doubt that it might contain an error which would invalidate the whole theory. I badly lacked colleagues who could check my work.Takagi spoke of his work on class field theory, building on Heinrich Weber's work, at the International Congress of Mathematicians in Strasbourg in 1920. While in Europe he visited Hecke and Blaschke in Hamburg. He wrote his most important paper in 1920 which introduced the Takagi class-field theory generalising Hilbert's class field. In 1922 Siegel persuaded Artin to read this paper and its significance was realised. It became the framework of algebraic number theory. Hasse included Takagi's theory in his treatise on class field theory a few years later. In 1925 Hilbert wrote to Takagi in Japan asking if his paper could be published in Mathematische Annalen.
Around this time other mathematicians working in the same area as Takagi started to be appointed to Tokyo University and at last he had the mathematical colleagues he had longed for. Iyanaga, the author of , became Takagi's student in 1926. He describes his teaching style :-
[Takagi gave] his lectures without prepared papers, showing however, traits of spirit from time to time with sharp critical remarks, sometimes mixed with jokes. He spoke rather slowly in a low voice and almost never repeated the same thing; he wrote very neatly on the blackboard but the colour of his chalk was rather light; the speed of flow of his lecture was quite rapid and the students had to listen with great attention.Soon honours began to be given to Takagi for his outstanding work. He was honoured by Czechoslovakia, the university of Oslo, and the National Research Council of Japan. Fueter was President of the International Congress of Mathematicians at Zürich in 1932 and Takagi was appointed Vice-President. He served on the committee to award the first Fields' Medals for the 1936 Congress.
In 1936 Takagi retired but continued publishing books and papers. His two most important books from this time are Introduction to analysis (1938), Algebraic number theory (1948) and an important work on the history of mathematics in the 19th century. Takagi continued to live in Tokyo after he retired until 1945 when his house was destroyed by bombing near the end of World War II. He returned to the village of his birth, coming back to Tokyo in 1947 to live with his eldest son. His wife died of cancer in 1952 and Takagi himself died at the age of 88 in the hospital of Tokyo university.
Article by: J J O'Connor and E F Robertson
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