Zur theorie der Ringklassenkörper über imaginär-quadratischen Zahlkörpern

AbstractLet Σ be an imaginary quadratic number field, and Ωf the ring class field extension of Σ for a natural number f as conductor. For the investigations of class number and unit group of the subfields of Ωf by means of the methods of complex multiplication one often uses the quotients Δ(a)Δ(b) of the singular values of the discriminant Δ occurring in the theory of modular functions. As is well known, they are contained in Ωf. But it is not known whether those quotients generate Ωf over Σ. Corollary 1 of this paper solves this problem. Moreover Theorems 2, 3 exhibit more general methods of generating the subfields by means of relative norms