Open Access

Dirk Hachenberger

• A characterization of normal bases and complete normal bases in GF(q^(r^n)) over GF(q), where q > 1 is any prime power, r is any prime number different from the characteristic of GF(q), and n >= 1 is any integer, leads to a general construction scheme of series (v_n)n>=0 in GF(q^(r^infinite) having the property that the partial Sums w_n are free or completely free in GF(q^(r^n)) over GF(q), depending on the choice of v_n. In the case where r is an odd prime divisor of q - 1 or where r = 2 and q = 1 mod 4, for any integer n >= 1, all free and completely free elements in GF(q^(r^n)) over GF(q) are explicitly determined in terms of certain roots of unity. In the case where r = 2 and q = 3 mod 4, for any n >= 1, in terms of certain roots of unity, an explicit recursive construction for free and completely free elements in GF(q^(2^n)) over GF(q) is given. As an example, for a particular series of completely free elements the corresponding minimal polynomials are given explicitly.