Normal Bases and Completely Free Elements in Prime Power Extensions over Finite Fields

We continue the work of the previous paper (Hachenberger,Finite Fields Appl., in press), and, generalizing some of the results obtained there, we give explicit constructions of free and completely free elements in GF(qrn) over GF(q), wherenis any nonnegative integer and whereris any odd prime number which does not divide the characteristic of GF(q) or wherer= 2 andq≡ 1 mod 4. Together with results on the case wherer= 2 andq≡ 3 mod 4 obtained in the previous paper and results on the well-known case whereris equal to the characteristic of GF(q), we are able to explicitly determine free and completely free elements in GF(qm) over GF(q) for every nonnegative integermand every prime powerq.