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