In 1844 and 1880, two former classmates recorded their liebsten Jugendträume (“dearest dreams of youth”) and sketched a path into “where the most difficult parts of analysis and number theory join”. Gotthold Eisenstein and Leopold Kronecker, both students of Peter Gustav Lejeune Dirichlet, had hoped that special complex functions could be used to generate various types of algebraic numbers. In 1900, David Hilbert named this as one of the 23 greatest unsolved problems in mathematics; more than a century later, this “dearest dream of youth” has only been partially realized. We will journey from milestone to milestone along their visionary path, look back at how far mathematics has progressed, and take a peek at what lies just ahead.