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

Abstract

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 ≡ 3 mod 4 and let m ( q ) ≥ 3 be the largest integer such that 2 m ( q ) divides q 2 − 1 ; if E = F q 2 l , 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 ) 2 l − 2 . We are further going to discuss lower bounds on the number of such elements in r -power extensions, where r = 2 and q ≡ 1 mod 4 , or where r is an odd prime, or where r is equal to the characteristic of the underlying field.