Richard Dedekind: Numbers and Ideals

Richard Dedekind (1831-1916) is among the nineteenth century mathematicians 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 continually 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 precisely 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.