Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analogues of the roots of unity, as complex numbers that are particular values of the exponential function; the requirement is that such numbers should generate a whole family of further number fields that are analogues of the cyclotomic fields and their subfields.

The classical theory of complex multiplication, now often known as the Kronecker Jugendtraum, does this for the case of any imaginary quadratic field, by using modular functions and elliptic functions chosen with a particular period lattice related to the field in question. Goro Shimura extended this to CM fields. In the special case of totally real fields, a solution was given by Dasgupta and Kakde. This provides an effective method to construct the maximal abelian extension of any totally real field. The method rests on p-adic integration and the solution it provides for totally real fields is different in nature from what Hilbert had in mind in his original formulation. A solution in the more special case of totally real quadratic fields, also resting on p-adic methods, was given by Darmon, Pozzi and Vonk.

The general case of Hilbert's 12th Problem is still open as of 2023.

Leopold Kronecker described the complex multiplication issue as his liebster Jugendtraum, or “dearest dream of his youth”.