The German mathematician Kronecker proved that all the Galois extensions of the rationals Q with Abelian Galois groups are subfields of cyclotomic fields Q(μ_n), where μ_n is the group of nth 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 K, 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 number theory.

imaginary quadratic field | j-function