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

Dirk Hachenberger

erschienen 01.01.2010 Discrete Mathematics 310 [Issue 22], 3246-3250

Summary: The present paper is a continuation of the author’s work [Glasg. Math. J. 43, No. 3, 383–398 (2001; Zbl 0996.11074)] on primitivity and complete normality. For certain 2-power extensions E over a Galois field Fq, we are going to establish the existence of a primitive element which simultaneously generates a normal basis over every intermediate field of E/Fq. The main result is as follows: Let q 3 mod 4 and let m(q) 3 be the largest integer such that 2m(q) divides q2 1; if E = Fq2l , where l m(q) + 3, then there exists a primitive element in E that is completely normal over Fq.

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−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.