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 < 1050 with p < 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.
I. H. Morgan and G. L. Mullen, "Completely Normal Primitive Basis Generators of Finite Fields," Utilitas Mathematica, Utilitas Mathematica Publishing Incorporated, Jan 1996.
Mathematics and Statistics
Article - Journal
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