Subgroups of HolQ8as Galois Groups Original Research Article

In this paper, we consider nonsplit Galois theoretical embedding problems with cyclic kernel of prime orderp, in the case where the ground field has characteristic ≠p. It is shown that such an embedding problem can always be reduced to another embedding problem, in which the ground field contains the primitivepth roots of unity, and the group extension is central. The reduction is effective, in the sense that a solution to the reduced embedding problem induces a solution to the original embedding problem and that all solutions to the original embedding problem are induced in this way from solutions to the reduced embedding problem. The simplest case of this reduction is then used to give criteria for the realisability of four subgroups of the holomorph HolQ8, whereQ8is the quaternion group of order 8, including the holomorph itself.