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Kronecker, Leopold

berlin theory complex research

(1823–91) German mathematician: developed algebraic number theory and invented the Kronecker delta.

Kronecker was born into a rich Jewish family and was taught at school by E E Kummer (1810–93), who became a lifelong friend. From this time sprang his interest in number theory and arithmetic, and he went on to take his degree at Berlin (1843), receiving his doctorate in 1845 for research on complex units. Kronecker spent the next 10 years managing the family estate and an uncle’s banking business: he prospered and married. When he returned to Berlin to do mathematics he was financially independent, and only lectured at the Berlin Academy from 1861 for his own pleasure. He declined the chair at Göttingen (1868) but accepted Kummer’s old chair at Berlin, holding it until his death. Kronecker was a man of wide culture, supporting the arts and interested in philosophy and Christian theology, he became a Christian shortly before his death.

In his early mathematical research on complex units Kronecker nearly anticipated Kummer’s famous concept of ideal numbers, and his work on number theory, algebra and elliptic functions unified much previous research. On algebraic numbers he rederived much of existing theory without referring to what he (incorrectly) claimed were the ill-defined complex and irrational numbers. He once made the comment in an after-dinner speech: ‘God made the integers, all else is the work of man’. He was often in debate with and G Cantor (1845–1918) and this gave rise to his system of axioms (1870) to support a formalist viewpoint. In linear algebra Kronecker invented the Kronecker delta (s mn = 1 if m = n and s mn = 0 otherwise), and established its use when evaluating determinants. It is famous as the first example of a tensor quantity being used.

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