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 extensions of an arbitrary number field $K$ in of values of transcendental functions.

Class field theory gives a solution to this problem in the case where $K=\mathbb{Q}$, the field of rational numbers. Specifically, the Kronecker-Weber theorem gives that any number field sits inside one of the cyclotomic fields $\mathbb{Q}({\zeta }_n)$ for some $n$. Refining this only slightly gives that we can explicitly generate all abelian extensions of $\mathbb{Q}$ by adjoining values of the transcendental function $e^{2\pi iz}$ for certain points $z\in \mathbb{Q}/\mathbb{Z}$.

A slightly more complicated example is when $K$ is a quadratic imaginary extension of $\mathbb{Q}$, 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 $j$-function (as in elliptic curves) and Weber’s $w$-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.