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.

**References**

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

© 1996-9

1999-05-25