Existence results on k-normal elements over finite fields

TL;DR

This paper investigates the existence of $k$-normal elements over finite fields, extending the concept of normal elements, and explores their properties, existence conditions, and connections to polynomial factorizations.

Contribution

It provides new general results on the existence of $k$-normal elements with properties like being primitive or having large order, linking them to polynomial factorizations.

Findings

Many existence results for $k$-normal elements with specific properties.

Connections established between $k$-normal elements and factorization of $x^n-1$ over finite fields.

Enhanced understanding of conditions for the existence of $k$-normal elements.

Abstract

An element $\alpha \in \mathbb {F}_{q^n}$ is normal over $\mathbb {F}_q$ if $\alpha$ and its conjugates $\alpha, \alpha^q, \cdots \alpha^{q^{n-1}}$ form a basis of $\mathbb {F}_{q^n}$ over $\mathbb {F}_q$. Recently, Huczynska, Mullen, Panario and Thomson (2013) introduce 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 $\mathbb {F}_q$.