Hilbert's 12th problem

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. The general case is still open as of 2014^[update]. Leopold Kronecker described the complex multiplication issue as his liebster Jugendtraum or “dearest dream of his youth”.

The fundamental problem of algebraic number theory is to describe the fields of algebraic numbers. The work of Galois made it clear that field extensions are controlled by certain groups, the Galois groups. The simplest situation, which is already at the boundary of what we can do, is when the group in question is abelian. All quadratic extensions, obtained by adjoining the roots of a quadratic polynomial, are abelian, and their study was commenced by Gauss. Another type of abelian extension of the field Q of rational numbers is given by adjoining the nth roots of unity, resulting in the cyclotomic fields. Already Gauss had shown that, in fact, every quadratic field is contained in a larger cyclotomic field. The Kronecker–Weber theorem shows that any finite abelian extension of Q is contained in a cyclotomic field. Kronecker's (and Hilbert's) question addresses the situation of a more general algebraic number field K: what are the algebraic numbers necessary to construct all abelian extensions of K? The complete answer to this question has been completely worked out only when K is an imaginary quadratic field or its generalization, a CM-field.

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