Kronecker proved that all the Galois extensions of the Rationals Q with Abelian Galois groups are Subfields of cyclotomic fields , where is the group of th Roots of Unity. He then sought to find a similar function whose division values would generate the Abelian extensions of an arbitrary Number Field. He discovered that the j-Function works for Imaginary quadratic number fields , but the completion of this problem, known as Kronecker's Jugendtraum (``dream of youth''), for other fields remains one of the great unsolved problems in Number Theory.
See also j-Function
Shimura, G. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton, NJ: Princeton University