Existence of primitive 2-normal elements in finite fields

Abstract

An element α∈Fqn is normal over Fq if B={α,αq,αq2,⋯,αqn−1} forms a basis of Fqn as a vector space over Fq. It is well known that α∈Fqn is normal over Fq if and only if gα(x)=αxn−1+αqxn−2+⋯+αqn−2x+αqn−1 and xn−1 are relatively prime over Fqn, that is, the degree of their greatest common divisor in Fqn[x] is 0. Using this equivalence, the notion of k-normal elements was introduced in Huczynska et al. (2013): an element α∈Fqn is k-normal over Fq if the greatest common divisor of the polynomials gα[x] and xn−1 in Fqn[x] has degree k; so an element which is normal in the usual sense is 0-normal.Huczynska et al. made the question about the pairs (n,k) for which there exist primitive k-normal elements in Fqn over Fq and they got a partial result for the case k=1, and later Reis and Thomson (2018) completed this case. The Primitive Normal Basis Theorem solves the case k=0. In this paper, we solve completely the case k=2 using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of k-normal elements in Fqn.