## Class field theory in characteristic $$p$$, its origin and development.(English)Zbl 1068.11073

Miyake, Katsuya (ed.), Class field theory – its centenary and prospect. Proceedings of the 7th MSJ International Research Institute of the Mathematical Society of Japan, Tokyo, Japan, June 3–12, 1998. Tokyo: Mathematical Society of Japan (ISBN 4-931469-11-6/hbk). Adv. Stud. Pure Math. 30, 549-631 (2001).
Summary: Today’s notion of “global field” comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new impetus was given by the seminal thesis of E. Artin (1921, published in 1924). In this exposition I shall report on the development during the twenties and thirties of our century, with emphasis on the emergence of class field theory for function fields. The names of F. K. Schmidt, H. Hasse, E. Witt, C. Chevalley (among others) are closely connected with that development.
For the entire collection see [Zbl 0968.00031].

### MSC:

 11R37 Class field theory 11-03 History of number theory 11R58 Arithmetic theory of algebraic function fields 01A60 History of mathematics in the 20th century