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The mathematical legacy of Kronecker is impressive. The list of mathematicians who took up his problems includes Adolph Hurwitz, David Hilbert, Kurt Hensel, Julius König, Ernst Steinitz, Erich Hecke, Helmut Hasse, Carl Ludwig Siegel, Hermann Weyl and, recently, André Weil, Robert Langlands, Harold Edwards. But, although nobody has ever doubted the brilliance of Kronecker's results and insights, it is relatively safe to say with Edwards that, more than a century after the publication of his complete works, "there are important passages in Kronecker's work that no one, ever has fully understood, other than Kronecker himself" (1987, p. 29). This situation is the result of a lack of interest caused by disdain for Kronecker's well-known foundational stance and for his style of mathematics. Indeed, Kronecker's mathematical practice features an insistence on providing algorithms which is alien to the abstract, axiomatic approach which has been so dominant in the past hundred years, in particular in Bourbaki's treatises. From this point of view, algebra consists in the study of (highly) abstract notions — the structures-mères — such as groups, rings, field, and so forth. Dedekind presented and refined his theory of ideals in successive versions of his Supplement XI to Lejeune-Dirichlet's Vorlesungen über Zahlentheorie (1879). Adopted by Hilbert in his Zahlbericht and studied carefully by emmy Noether and her students, Dedekind's ideals played a crucial role in the development of modern algebraic number theory.

DOI: 10.1007/978-94-009-1575-6_11

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