Normal bases and completely free elements in prime power extensions over finite fields (Q1908793)

For \(q\) any prime power, \(m\) any positive integer, an element \(w\) of the Galois field \(GF(q^m)\) is free (resp., completely free) over \(GF(q)\) provided it generates a normal basis in \(GF(q^m)\) over \(GF(q)\) (resp., it simultaneously generates normal bases in \(GF(q^m)\) over each intermediate field \(GF(q^d)\) of \(GF(q^m)\) over \(GF(q))\). Continuing his earlier work [Finite Fields Appl. 2, No. 1, 1-20 (1996; Zbl 0848.11061)], the author is able to explicitly determine free and completely free elements in \(GF(q^m)\) over \(GF(q)\) for every nonnegative integer \(m\) and every prime power \(q\). Moreover, in many cases he is able to describe all completely free elements in terms of a certain primitive root of unity.