Galois Module Structure and Elliptic Functions Original Research Article

Let K be a quadratic imaginary number field and Rf the ring class field module f over K, f set membership, variant image. Let imagef denote the order of conductor f in K and let image*, image be proper imagef-ideals such that image*2 subset of or equal to image subset of or equal to image*. Let τ denote the Weber function and let image denote an auxiliary imagef-ideal. For certain extensions Rf(τ(1 imageimage))/Rf(τ(1 imageimage*)) it is shown that the ring of integers in Rf(τ(1 imageimage)) is a free rank one module over the associated order of Rf(τ(1 imageimage))/Rf(τ(1 imageimage*)).