The German mathematician 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 fields , but the completion of this problem, known as Kronecker's Jugendtraum ("dream of youth"), for more general fields remains one of the great unsolved problems in [/u]number theory[/u].