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) ) on primitivity and complete normality. For certain 2-power extensions over a Galois field , we are going to establish the existence of a primitive element which simultaneously generates a normal basis over every intermediate field of . The main result is as follows: Let and let be the largest integer such that divides ; if , where , then there exists a primitive element in that is completely normal over . 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 . We are further going to discuss lower bounds on the number of such elements in -power extensions, where and , or where is an odd prime, or where is equal to the characteristic of the underlying field. [Copyright &y& Elsevier]