A Unified Differential-Algebraic Approach to Class Number One Problems for Quadratic and General Number Fields via Finite Representations, Heights, and Branch Index Systems

This paper systematically applies the theories of differential-algebraic closures and finite representations, combined with a newly established equivalence theorem between differential-algebraic heights and classical heights, to present a novel unified framework for the class number one problems of imaginary quadratic fields Q (√−d) and real quadratic fields Q (√d). For imaginary quadratic fields, the core idea is to parametrize the analytic quantity L (1, χ−d) from the class number formula as a differential algebraic representation of elliptic functions, thereby transforming the class number one condition into a system of consistency equations concerning branch indices. Through a deep analysis of the solvability of this system modulo n and its 2-adic properties, we rigorously prove that the class number is one only for d ∈ 1, 2, 3, 7, 11, 19, 43, 67, 163. For real quadratic fields, we realize the fundamental unit ϵd and L (1, χd) within a differential-algebraic closure via finite representations and similarly establish a branch index system. Utilizing Baker’s theorem to analyze the compatibility conditions of this system, we prove that there are only finitely many real quadratic fields with class number one. The theory of differential algebraic height, which permeates the entire paper, not only provides a profound height characterization for the class number one of both types of fields (height 2 for imaginary quadratic fields, height 3 for real quadratic fields) but also offers a unified intrinsic explanation for the finiteness of class number problems. This paper not only reproves classical results but also demonstrates the profound application and potential of differential-algebraic methods at the intersection of analytic and transcendental number theory. Furthermore, we extend the framework to higher-dimensional settings, establishing a multivariate differential-algebraic finite representation theory, presenting branch index systems with combinatorial correction terms, and proving a finiteness theorem for the class number one problem of general number fields. This unifies and extends traditional results to a new theoretical level. Methodological Innovations: The key innovations of this work are threefold: (1) the introduction and systematic development of the concept of differential-algebraic height hDA as a measure of representational complexity, rigorously shown to be equivalent to the classical Weil height; (2) the constructive proof of the differential-algebraic finite representation theorem for solutions of linear differential equations with solvable Galois groups, which provides an explicit parametrization via branch indices sm; (3) the translation of class number conditions into branch index consistency systems—a set of linear (or polynomial) equations modulo n—whose solvability, analyzed via p-adic methods (for imaginary quadratic) and Baker-type transcendence bounds (for real and general fields), yields the desired finiteness or classification. These components form a cohesive, self-contained framework that transforms analytic number theory problems into combinatorial-algebraic ones.