Kronecker’s Jugendtraum (Jugendtraum is German for “youthful dream”) describes a central problem in class field theory, to explicitly describe the abelian extensions of an arbitrary number field in of values of transcendental functions.
Class field theory gives a solution to this problem in the case where , the field of rational numbers. Specifically, the Kronecker-Weber theorem gives that any number field sits inside one of the cyclotomic fields for some . Refining this only slightly gives that we can explicitly generate all abelian extensions of by adjoining values of the transcendental function for certain points .
A slightly more complicated example is when is a quadratic imaginary extension of , in which case Kronecker’s Jugendtraum has been solved by the theory of “complex multiplication” (see CM-field). The specific transcendental functions which generate all these abelian extensions are the -function (as in elliptic curves) and Weber’s -function.
Though there are partial results in the cases of CM-fields or real quadratic fields, the problem is largely still open (http://planetmath.org/OpenQuestion), and earned great prestige by being included as Hilbert’s twelfth problem.
|Date of creation||2013-03-22 15:01:08|
|Last modified on||2013-03-22 15:01:08|
|Last modified by||mathcam (2727)|