# A note on primitive $1-$normal elements over finite fields

## TL;DR

This paper extends the existence results of primitive 1-normal elements over finite fields to include the previously excluded case where q=2, broadening the understanding of such elements.

## Contribution

It generalizes the existence theorem for primitive 1-normal elements to the case q=2, which was not covered in earlier work.

## Key findings

- Existence of primitive 1-normal elements for q=2 established.
- Extends previous results to include the case q=2.
- Provides a more complete understanding of primitive 1-normal elements.

## Abstract

Let $q$ be a prime power of a prime $p$, $n$ a positive integer and $\mathbb F_{q^n}$ the finite field with $q^n$ elements. The $k-$normal elements over finite fields were introduced and characterized by Huczynska et al (2013). Under the condition that $n$ is not divisible by $p$, they obtained an existence result on primitive $1-$normal elements of $\mathbb F_{q^n}$ over $\mathbb F_q$ for $q>2$. In this note, we extend their result to the excluded case $q=2$.