Normal bases and completely free elements in prime power extensions over finite fields
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.
prime power extensions over finite fields
scientific article; zbMATH DE number 851859