Four non-abelian groups of order p4 as Galois groups

Let p be an odd prime and let F be arbitrary field of characteristic not p, containing a primitive pth root of unity ζ. In this paper, we prove a criterion, giving the obstructions to realizability of p-groups as Galois groups over F, having a factor-group of the kind H×Cp. We apply this to the non-abelian groups of orders p3 and p4. Where it is possible, we give a description of all Galois extensions realizing these groups. We discuss also automatic realizations and local fields.