Existence of primitive 1-normal elements in finite fields

Abstract

An element α∈Fqn is normal if B={α,αq,…,αqn−1} forms a basis of Fqn as a vector space over Fq; in this case, B is a normal basis of Fqn over Fq. The notion of k-normal elements was introduced in Huczynska et al. (2013) [10]. Using the same notation as before, α is k-normal if B spans a co-dimension k subspace of Fqn. It can be shown that 1-normal elements always exist in Fqn, and Huczynska et al. (2013) [10] show that elements that are simultaneously primitive and 1-normal exist for q≥3 and for large enough n when gcd⁡(n,q)=1 (we note that primitive 1-normals cannot exist when n=2). In this paper, we complete this theorem and show that primitive, 1-normal elements of Fqn over Fq exist for all prime powers q and all integers n≥3, thus solving Problem 6.3 from Huczynska et al. (2013) [10].