Chapter 2 Richard Dedekind: Numbers and Ideals Richard Dedekind (1831-1916) is among the nineteenth century mathemati- cians whose work has been most extensively studied and praised. 1 No one cared to stress the importance of his contributions, and to extol the pioneering character of his approach, more emphatically than Emmy Noether. She con- tinually advised her students to read and re-read Dedekind's works, in which she saw an inexhaustible source of inspiration. When praised for her own innovations, she used to repeat: "Es steht alles schon bei Dedekind.,,2 The close links between the mathematical ideas of these two masters, Dedekind and Noether, manifest themselves in their respective works on the theory of ideals more evidently than in any other context. Yet at the same time, it is pre- cisely by examining their respective works in this field that one can best define the significant differences between their approaches and thus explain the sense in which Noether's work may be said to be more "structural" than Dedekind's. Understanding these differences enables us to identify the meaning of the change that the images of algebra, and algebraic research at large, underwent between the last third of the nineteenth century and the 1920s. Dedekind studied in his native city of Braunschweig and in Gbttingen. In 1852 he completed his doctoral dissertation, working under the supervision of Gauss. Later in 1854, he habilitated with Bernhard Riemann (1826-1866).3 During his first years as Privatdozent in Gbttingen he worked in close collab- oration with Peter Lejeune Dirichlet (1805-1859). Thus, the decisive influence of the leading mathematicians of the early Gbttingen tradition marked Dede- kind's formative years. This influence was clearly manifest in all of his later I. Cf. Biennann 1971; Dieudonne (ed.) 1978, Vol. 1,200-214; Dugac 1976; Edwards 1980, 1983; Ferreir6s 1993; Ferreir6s 1999,81-113 & 215-256; Fuchs 1982; Hawkins 1971, 1974; Haubrich 1988; Mehrtens 1979 (Chpt. 2), 1979a, 1982; Noether and Cavailles (eds.) 1937; Pia i Car- rera 1993; Purkert 1971, 1976; Scharlau 1981, 1982; Zincke 1916. 2. Cf. Dedekind 1964, iv. L. Corry, Modern Algebra and the Rise of Mathematical Structures © Birkhäuser Verlag 2004