ON SOME SUBGROUPS OF THE IDEAL CLASS GROUP OF REAL QUADRATIC ORDERS

Abstract

Let Δ > 0 be a quadratic discriminant and Cl_∆ be the ideal class group of the quadratic order 𝒪_∆. In this paper, we introduce two subgroups Cl_∆⁽¹⁾ and Cl_∆⁽²⁾ of Cl_∆ and, by using them, give a necessary and sufficient condition for h_∆ = 1 to hold. This corresponds to a generalization of a theorem of Louboutin. Moreover, we determine an explicit form of generators of Cl_∆⁽²⁾.