Title: 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 $q=2$.