Jugendtraum

Kronecker proved that all the Galois extensions of the Rationals Q with Abelian Galois groups are Subfields of cyclotomic fields $Q(\mu_n)$, where $\mu_n$ is the group of $n$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 $K$, 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

References

Shimura, G. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton, NJ: Princeton University Press, 1981.