Realizable Classes of Quaternion Extensions of Degree 4ℓ Original Research Article

Let k be a number field and Ok its ring of integers. Let Γ be the generalized quaternion group of order 4ℓ, where ℓ is an odd prime number. Let image be a maximal Ok-order in k[Γ] containing Ok[Γ] and imageℓ(image) its class group. We denote by image(image) the subset of imageℓ(image) formed by the realizable classes in the sense of McCulloh. In this article we define two subsets of image(image) and show that they are subgroups of imageℓ(image) provided that 2 and ℓ are unramified in k/image.