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# Kronecker

[kroh-nek-er; Ger. kroh-nek-uhr]
Kronecker, Leopold, 1823-91, German mathematician. After making a fortune in business he devoted his attention to mathematics and became professor at the Univ. of Berlin in 1883. Noted as an algebraist, he was a pioneer in the field of algebraic numbers and in formulating the relationship between the theory of numbers, the theory of equations, and elliptic functions.
In mathematics, Kronecker's theorem, named after Leopold Kronecker, is a result in diophantine approximations applying to several real numbers xi, for 1 ≤ iN, that generalises the equidistribution theorem, which implies that an infinite cyclic subgroup of the unit circle group is a dense subset. In terms of physical systems, it has the consequence that planets in circular orbits moving uniformly around a star will, over time, assume all alignments, unless there is an exact dependency between their orbital periods.

In the case of N numbers, taken as a single N-tuple and point P of the torus

T = RN/ZN,

the closure of the subgroup <P> generated by P will be finite, or some torus T′ contained in T. The original Kronecker's theorem (Leopold Kronecker, 1884) stated that the necessary condition for

T′ = T,

which is that the numbers xi together with 1 should be linearly independent over the rational numbers, is also sufficient. Here it is easy to see that if some linear combination of the xi and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a character χ of the group T other than the trivial character takes the value 1 on P. By Pontryagin duality we have T′ contained in the kernel of χ, and therefore not equal to T.

In fact a thorough use of Pontryagin duality here shows that the whole Kronecker theorem describes the closure of <P> as the intersection of the kernels of the χ with

χ(P) = 1.

This gives an (antitone) Galois connection between monogenic closed subgroups of T (those with a single generator, in the topological sense), and sets of characters with kernel containing a given point. Not all closed subgroups occur as monogenic; for example a subgroup that has a torus of dimension ≥ 1 as connected component of the identity element, and that is not connected, cannot be such a subgroup.

The theorem leaves open the question of how well (uniformly) the multiples mP of P fill up the closure. In the one-dimensional case, the distribution is uniform by the equidistribution theorem.