Kronecker’s Jugendtraum

Kronecker’s Jugendtraum (Jugendtraum is German for “youthful dream”) describes a central problem in class field theory, to explicitly describe the abelian extensionsMathworldPlanetmath of an arbitrary number fieldMathworldPlanetmath K in of values of transcendental functionsMathworldPlanetmath.

Class field theory gives a solution to this problem in the case where K=, the field of rational numbers. Specifically, the Kronecker-Weber theoremMathworldPlanetmath gives that any number field sits inside one of the cyclotomic fieldsMathworldPlanetmath (ζn) for some n. Refining this only slightly gives that we can explicitly generate all abelian extensions of by adjoining values of the transcendental function e2πiz for certain points z/.

A slightly more complicated example is when K is a quadratic imaginary extension of , in which case Kronecker’s Jugendtraum has been solved by the theory of “complex multiplicationMathworldPlanetmath” (see CM-field). The specific transcendental functions which generate all these abelian extensions are the j-functionMathworldPlanetmath (as in elliptic curves) and Weber’s w-function.

Though there are partial results in the cases of CM-fields or real quadratic fieldsMathworldPlanetmath, the problem is largely still open (, and earned great prestige by being included as Hilbert’s twelfth problem.

Title Kronecker’s Jugendtraum
Canonical name KroneckersJugendtraum
Date of creation 2013-03-22 15:01:08
Last modified on 2013-03-22 15:01:08
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 11R37