# Leopold Kronecker

**Leopold Kronecker** (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on "whole numbers", saying, "*God made the integers; all else is the work of man*" (Bell 1986, p. 477). This put Kronecker in opposition to some of the mathematical extensions of Georg Cantor. Kronecker was a student and lifelong friend of Ernst Kummer.

Kronecker wrote 1845 his dissertation, at the University of Berlin, on number theory, giving special formulation to units in certain algebraic number fields. Peter Gustav Dirichlet was his teacher.

After obtaining his degree, however, Kronecker managed the estate and business of his uncle, producing nothing mathematical for eight years. In his 1853 memoir on the algebraic solvability of equations, Kronecker extended the work of Évariste Galois on the theory of equations. He accepted a professorship at the University of Berlin in 1883.

Kronecker also contributed to the concept of continuity, reconstructing the form of irrational numbers in real numbers. In analysis, Kronecker rejected the formulation of a continuous, nowhere differentiable function by his colleague, Karl Weierstrass. In his 1850 paper, *On the Solution of the General Equation of the Fifth Degree*, Kronecker solved the quintic equation by applying group theory.

Kronecker's finitism made him a forerunner of intuitionism in foundations of mathematics.

The Kronecker delta and Kronecker product are named after Kronecker, as are the Kronecker-Weber theorem and Kronecker's theorem in number theory.

## References

- Eric Temple Bell,
*Men of Mathematics*, Simon and Schuster, New York, 1986.

## External link

- O'Connor, John J., and Edmund F. Robertson. "Leopold Kronecker".
*MacTutor History of Mathematics archive*.

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