Abstract: Göttingen, the mathematical town of the 19th century, witnessed not only the birth of Dedekind’s ideal theory, but also the origin of Riemann surfaces. Dedekind combined ideal theory with the analytic structure of Riemann surfaces, thereby enabling a more precise characterization of the properties of algebraic curves and advancing the algebraization of Riemann surfaces. This integration not only deepened the understanding of Riemann surfaces but also laid the foundation for the development of algebraic geometry. This paper interprets the connection between Dedekind’s ideal theory and Riemann surfaces through a close and meticulous analysis of the original literature, and examines its contribution to the algebraisation of Riemann surfaces. It is elucidated that Dedekind’s conceptualism and logicism provided a solid theoretical foundation and methodological guidance for the development of Riemann surfaces, revealing the intrinsic connections and philosophical significance underlying the development of mathematical theory.

Key Words: Ideal theory; Riemann surface; Theory of function; Richard Dedekind