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

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 $q=2$.

4 pages