Explicit Iterative Constructions of Normal Bases and Completely Free Elements in Finite Fields

A characterization of normal bases and complete normal bases in GF(qrn) over GF(q), whereq> 1 is any prime power,ris any prime number different from the characteristic of GF(q), andn≥ 1 is any integer, leads to a general construction scheme of series (vn)n≥0in GF(qr∞) ni≥0GF( qrn) having the property that the partial sumswn Σni 0 viare free or completely free in GF(qrn) over GF(q), depending on the choice ofvn. In the case whereris an odd prime divisor ofq− 1 or wherer= 2 andq≡ 1 mod 4, for any integern≥ 1, all free and completely free elements in GF(qrn) over GF(q) are explicitly determined in terms of certain roots of unity. In the case wherer= 2 andq≡ 3 mod 4, for anyn≥ 1, in terms of certain roots of unity, an explicit recursive construction for free and completely free elements in GF(q2n) over GF(q) is given. As an example, for a particular series of completely free elements the corresponding minimal polynomials are given explicitly.