du
, S. G. 1991. Kronecker’s Jugendtraum and Modular Functions. New York: Gordon & Breach.b. Liegnitz, Silesia, today Poland, 1823; d. Berlin, 1891
Algebra; number theory
One of the dominant mathematicians of the second half of the nineteenth century, Kronecker is best known today for his constructivist views and his contributions to number theory. After finishing his Ph.D. under the supervision of DIRICHLET [VI.36] in 1845, Kronecker left Berlin and mathematics in order to manage a family estate and to wind up his father-in-law’s banking business. These activities left him wealthy and free to return to Berlin and concentrate on mathematics without holding an academic position. In 1855, Kronecker’s former school teacher and closest scientific friend, ERNST EDUARD KUMMER [VI.40], also came to Berlin and stayed there until his death in 1893. In 1861, Kronecker became a member of the Berlin Academy of Sciences and started teaching courses at Berlin University. Kronecker valued the exchange with his Berlin colleagues (especially Kummer and WEIERSTRASS [VI.44]) highly, until a quarrel arose between Kronecker and Weierstrass in the 1870s, which drove Weierstrass to bitter, even anti-Semitic, complaints to others about Kronecker. After Kummer’s retirement in 1883, Kronecker occupied Kummer’s chair and stepped up his teaching activities as well as the frequency of his publications. This last active period was cut short when he died, shortly after the death of his wife.
Kronecker was renowned for the originality of his mathematical insight and became increasingly influential through the 1860s and 1870s. In 1868, he was offered the chair at Göttingen formerly held by GAUSS [VI.26], and was elected to the Paris Academy. After the Franco-Prussian war of 1870-71, he was invited to recommend mathematicians for the newly opened German university in Strasbourg; and in 1880 he became the managing editor of the Journal für die reine und angewandte Mathematik (otherwise known as Crelle’s Journal). He was often criticized for incomplete, unpublished, or incomprehensible proofs—JORDAN [VI.52] spoke of his colleagues’ “envy and despair” with regard to his results. Only in his later years was he explicit about his constructivist methodology. This constituted at least part of the quarrel with Weierstrass, and later prompted HILBERT [VI.63] to call Kronecker a “Verbotsdiktator” (“forbidding dictator”). Generally affable and hospitable, Kronecker was tough in defending his mathematical ideas and his claims to priority.
In his first works on solvable algebraic equations (in the early 1850s), he claimed not only the so-called Kronecker-Weber theorem (in today’s formulation: every finite Galois extension of the rational numbers with Abelian GALOIS GROUP [V.21] lies in a field generated by roots of unity; the first correct proof of it was given by Hilbert in 1896), but also a generalization to Abelian extensions of imaginary quadratic fields, which he later called his “liebster Jugendtraum” (“dearest dream of his youth”). This dream, which was incorrectly translated by Hilbert into his twelfth problem in 1900, is today part of CLASS FIELD THEORY [V.28] and the theory of complex multiplication. Such connections between algebra, analysis, and arithmetic continued to pervade Kronecker’s later work. Important results of Kronecker include class number relations and limit formulas in the theory of ELLIPTIC FUNCTIONS [V.31], the structure theorem for finitely generated Abelian groups, and a theory of bilinear forms.
In the late 1850s, Kronecker began to work on algebraic number theory, but only in 1881 did he publish his “Grundzüge einer arithmetischen Theorie der algebraischen Grössen,” dedicated to Kummer on the fiftieth anniversary of his doctorate. This mathematical testament contains an (incomplete) exposition of a unified arithmetical theory of algebraic numbers and algebraic functions. As a research program, it adumbrates important aspects of class field theory as well as of an arithmetico-geometric theory in dimensions higher than one. Kronecker’s concept of “divisor” is equivalent to Dedekind’s notion of “ideal” in the case of Dedekind domains, but is more restricted in the general case. Several mathematicians, such as H. Weber, K. Hensel, and G. König, took up the “Grundzüge” in their own work.
On a more general level, Kronecker asked for the complete arithmetization of pure mathematics, i.e., for the effective finitary reduction of pure mathematics to the notion of positive integer. For this, he propagated the introduction of indeterminates and equivalence relations, a method which he traced back to Gauss. In the case of a finite extension of the rational numbers, for instance, Kronecker is explicitly working with polynomials modulo an irreducible equation f (x) = 0, rather than adjoining a root of it.
Kronecker, L. 1895-1930. Werke, five volumes. Leipzig: Teubner.
Vldu, S. G. 1991. Kronecker’s Jugendtraum and Modular Functions. New York: Gordon & Breach.