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 F_qn. Such an element α ∈ F_qn simultaneously generates a normal basis of F_qn over all subfields F_qd where d divides n. In addition, α multiplicatively generates the group of all nonzero elements of F_qn. For each pⁿ < 10⁵⁰ with p < 97 a prime, we provide a completely normal primitive polynomial of degree n of minimal weight over the field F_p. Any root of such a polynomial will generate a completely normal primitive basis of F_pn over F_p. We have also conjectured a refinement of the primitive normal basis theorem for finite fields and, in addition, we raise several open problems.