On the discovery of the analogy between number and function fields: The origin of Dedekind rings. (Die Entdeckung der Analogie zwischen Zahl- und Funktionenkörpern: Der Ursprung der “Dedekind-Ringe”.) (German) Zbl 0943.11002

Ullrich, P.

Jahresber. Dtsch. Math.-Ver. 101, No. 3, 116-134 (1999).

The deep analogy between number fields and function fields was a source of deep problems and great insights especially in the second half of the 20th century. In this paper, the author restricts his attention to function fields of one variable over the field of complex numbers, and he describes how this analogy was discovered by Kronecker, Weierstraß, Dedekind and Weber, how mathematicians reacted to the introduction of algebraic methods to the theory of functions, and how Emmi Noether eventually formulated the axioms for Dedekind rings in the 1920s.
Reviewer’s remark: The term “Dedekind ring”, credited in this paper to I. S. Cohen in 1950, was already used by J. Dieudonné in 1947 [see Ann. Sci. Éc. Norm. Supér. (3) 64, 101–117 (1947; Zbl 0033.24801)].

Reviewer: Franz Lemmermeyer (Bonn)

MSC:

11-03	History of number theory
11R58	Arithmetic theory of algebraic function fields
01A55	History of mathematics in the 19th century
01A60	History of mathematics in the 20th century
11Rxx	Algebraic number theory: global fields
13F05	Dedekind, Prüfer, Krull and Mori rings and their generalizations
14-03	History of algebraic geometry
14H05	Algebraic functions and function fields in algebraic geometry

Keywords:

function fields; Kronecker; Dedekind; Weber; Hensel; Noether; Dedekind rings