Existence results on $k$-normal elements over finite fields

Abstract

An element \alpha \in \mathbb{F}_{q^n} is normal over \mathbb{F}_q if \alpha and its conjugates \alpha, \alpha^q, \dots, \alpha^{q^{n-1}} form a basis of \mathbb{F}_{q^n} over \mathbb{F}_q . In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of k -normal elements, generalizing the normal elements. In the past few years, many questions concerning the existence and number of k -normal elements with specified properties have been proposed. In this paper, we discuss some of these questions and, in particular, we provide many general results on the existence of k -normal elements with additional properties like being primitive or having large multiplicative order. We also discuss the existence and construction of k -normal elements in finite fields, providing a connection between k -normal elements and the factorization of x^n-1 over \mathbb{F}_q .