Primitive complete normal bases: Existence in certain 2-power extensions and lower bounds

The present paper is a continuation of the author's work (Hachenberger (2001) [3]) on primitivity and complete normality. For certain 2-power extensions E over a Galois field F-q, we are going to establish the existence of a primitive element which simultaneously generates a normal basis over every intermediate field of E/F-q. The main result is as follows: Let q equivalent to 3 mod 4 and let m(q) >= 3 be the largest integer such that 2(m(q)) divides q(2) - 1; if E = F-q21, where l >= m(q) + 3, then there exists a primitive element in E that is completely normal over F-q. Our method not only shows existence but also gives a fairly large lower bound on the number of primitive completely normal elements. In the above case this number is at least 4. (q - 1)(21-2). We are further going to discuss lower bounds on the number of such elements in r-power extensions, where r = 2 and q equivalent to 1 mod 4, or where r is an odd prime, or where r is equal to the characteristic of the underlying field. (C) 2010 Elsevier B.V. All rights reserved.