**Takagi** worked on class field theory, building on Heinrich Weber's work.

- Takagi studied Algebra for beginners by Todhunter and Geometry by Wilson.
- In 1891 Takagi began the third stage of his schooling which he took at the Third High School in Kyoto.
- Takagi therefore, after showing great talents at middle school, made the natural progression to Kyoto where he studied for three years.
- At Tokyo University Takagi took courses on calculus and analytic geometry.
- He eagerly read Heinrich Weber's Algebra text when it arrived in Japan and by 1898 Takagi had published his first paper.
- 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.
- Takagi wrote to Hilbert who arranged accommodation for him in Göttingen in a house in which he himself had previously lived.
- 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.
- On his return to Tokyo in 1903 Takagi proved a conjecture on abelian extensions of imaginary number fields made by Kronecker.
- Although Takagi was enthusiastic about research he did not continue to develop the work that he had begun in his thesis.
- 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.
- 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.
- He wrote his most important paper in 1920 which introduced the Takagi class-field theory generalising Hilbert's class field.
- 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.
- Soon honours began to be given to Takagi for his outstanding work.
- Fueter was President of the International Congress of Mathematicians at Zürich in 1932 and Takagi was appointed Vice-President.
- In 1936 Takagi retired but continued publishing books and papers.
- 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.

Born 21 April 1875, Kazuya Village (near Gifu), Japan. Died 29 February 1960, Tokyo, Japan.

View full biography at MacTutor

Origin Japan

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive