# Kronecker theorem

Given , , and ; then for any there exist integers , , and , , such that if and only if for any such that the number is also an integer. This theorem was first proved in 1884 by L. Kronecker (see ).

Kronecker's theorem is a special case of the following theorem , which describes the closure of the subgroup of the torus generated by the elements , : The closure is precisely the set of all classes such that, for any numbers with one has also (Cf. .) Under the assumptions of Kronecker's theorem, this closure is simply . This means that the subgroup of all elements of the form where , is dense in , while the subgroup of vectors where , is dense in . Kronecker's theorem can be derived from the duality theory for commutative topological groups (cf. Topological group), .

In the case , Kronecker's theorem becomes the following proposition: A class , where , generates as a topological group if and only if the numbers are linearly independent over the field of rational numbers. In particular, the torus as a topological group is monothetic, i.e. is generated by a single element.