Based on first principles of differential algebra, we construct for any number field K afinite-type differential system EK whose coefficients are completely determined by the arithmetic-topological data of K. We prove that the field of differential constants CK generated by the formal power series solutions of EK at a base point is canonically isomorphic to the maximal abelian extension Kab of K, and this isomorphism can be computed fully explicitly. Classical theories such as complex multiplication, Shimura varieties, function field class field theory, higher-dimensional local fields, Langlands correspondence, model-theoretic properties, and dynamical systems theory all emerge naturally as special cases or corollaries of the present theory. This paper not only provides a complete answer to Hilbert’s 12th problem,but also establishes a differential-algebraic foundation for class field theory, realizing a fundamental shift from “existence proofs” to “constructive algorithms”, and provides a unified framework for infinite-dimensional generalizations, quantization,and practical applications.
shifa liu (Wed,) studied this question.