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 EE over a Galois field FqFq, we are going to establish the existence of a primitive element which simultaneously generates a normal basis over every intermediate field of E/FqE/Fq. The main result is as follows: Let  q≡3mod4q≡3mod4and let  m(q)≥3m(q)≥3be the largest integer such that  2m(q)2m(q)divides  q2−1q2−1; if  E=Fq2lE=Fq2l, where  l≥m(q)+3l≥m(q)+3, then there exists a primitive element in  EEthat is completely normal over  FqFq.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)2l−24⋅(q−1)2l−2. We are further going to discuss lower bounds on the number of such elements in rr-power extensions, where r=2r=2 and q≡1mod4q≡1mod4, or where rr is an odd prime, or where rr is equal to the characteristic of the underlying field.