Ullrich, P. 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 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) Cited in 1 Document 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 Citations:Zbl 0033.24801 PDF BibTeX XML Cite \textit{P. Ullrich}, Jahresber. Dtsch. Math.-Ver. 101, No. 3, 116--134 (1999; Zbl 0943.11002) OpenURL