Completely Normal Primitive Basis Generators of Finite Fields

For q a prime power and n ≥ 2 an integer we consider the existence of completely normal primitive elements in finite field Fqn. Such an element α ∈ Fqn simultaneously generates a normal basis of Fqn over all subfields Fqd where d divides n. in addition, α multiplicatively generates the group of all nonzero elements of Fqn. For each pn \u3c 1050 with p \u3c 97 a prime, we provide a completely normal primitive polynomial of degree n of minimal weight over the field Fp. Any root of such a polynomial will generate a completely normal primitive basis of Fpn over Fp. We have also conjectured a refinement of the primitive normal basis theorem for finite fields and, in addition, we raise several open problems