Teiji Takagi one of the leading number theorists of this century, is most renowned as the founder of class field theory. This volume reflects the stages of his development of this theory. Inspired by a genial idea related to analytic number theory, he developed a beautiful general theory of abelian extensions of algebraic number fields which he addressed at the ICM 1920 at Strasbourg. This report ends with a problem to generalize the results to the case of normal, not necessarily abelian extensions. Up to now this problem has stimulated research. This second edition incorporates the whole contents of "The Collected Papers of Teiji Takagi" edited by S. Kuroda, published by Iwanami Shoten in 1974. Following additions have been made: Note on Eulerian squares (1946).- Concept of numbers.- K. Iwasawa: On arithmetical papers of Takagi.- K. Yosida: On analytical papers of Takagi.- S. Iyanaga: On life and works of Takagi.

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Teiji Takagi, one of the leading number theorists of the twentieth century, is most renowned as the founder of class field theory. This volume reflects the stages of his development of this theory. Inspired by a genius idea related to analytic number theory, he developed a beautiful general theory of abelian extensions of algebraic number fields, which he addressed at the ICM 1920 at Strasbourg. That report ended with a problem on generalizing the results to the case of normal, not necessarily abelian, extensions. Up to the present time, this problem has stimulated research. This second edition incorporates the entire contents of The Collected Papers of Teiji Takagi, edited by S. Kuroda and published by Iwanami Shoten in 1974, and includes the following additions: “Note on Eulerean squares” (T. Takagi, 1946), “On the concept of numbers” (T. Takagi), “On papers of Takagi in number theory” (K. Iwasawa), “On papers of Takagi in analysis” (K. Yosida), and “On the life and works of Teiji Takagi” (S. Iyanaga).

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