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 number theory.
Jugendtraum
See also
Imaginary Quadratic Field, j-FunctionExplore with Wolfram|Alpha
References
Shimura, G. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton, NJ: Princeton University Press, 1981.Referenced on Wolfram|Alpha
JugendtraumCite this as:
Weisstein, Eric W. "Jugendtraum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Jugendtraum.html