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 $\mathbb{F}_q$.