On $k$-normal elements over finite fields

TL;DR

This paper explores the construction and existence of $k$-normal elements over finite fields, providing new methods and proving the existence of primitive $k$-normal elements for many cases, especially when $k$ is within a certain range.

Contribution

The paper introduces alternative constructions for $k$-normal elements and establishes a sieve inequality for their primitive variants, expanding known existence results.

Findings

Existence of primitive $k$-normal elements for many $k$ in field extensions.

New sieve inequality for primitive $k$-normal elements.

Proof of primitive $k$-normal elements in $ ext{F}_{q^n}$ for $k$ in $[1, n/4]$ with certain conditions.

Abstract

The so called $k$-normal elements appear in the literature as a generalization of normal elements over finite fields. Recently, questions concerning the construction of $k$-normal elements and the existence of $k$-normal elements that are also primitive have attracted attention from many authors. In this paper we give alternative constructions of $k$-normal elements and, in particular, we obtain a sieve inequality for the existence of primitive, $k$-normal elements. As an application, we show the existence of primitive $k$-normal elements for a significant proportion of $k$'s in many field extensions. In particular, we prove that there exist primitive $k$-normals in $\mathbb F_{q^n}$ over $\mathbb F_q$ in the case when $k$ lies in the interval $[1, n/4]$, $n$ has a special property and $q, n\ge 420$.