Deconstruction of a Dirichlet–Nazimow formula

Abstract

Dirichlet (J Reine Angew Math 24, 1842) announced a generalization of his class number formula for real quadratic fields to biquadratic fields containing Q(i ), by replacing the circular trigonometric functions with certain elliptic trigonometric functions. Subsequently, Nazimow (Ann Sci École Norm Sup (3) 5:147-176, 1888) published the corresponding formula. Here we analyze the Dirichlet-Nazimow formula and update its proof in the context of Class Field Theory.