Existence of primitive k-normal elements for critical values over finite fields

Let ${\mathbb{F}}_{q^n}$ be a finite field with $q^n$ elements. An element $\alpha \in {\mathbb{F}}_{q^n}$ is called $k$-normal over ${\mathbb{F}}_q$ if $\alpha$ and its conjugates generate a vector subspace of ${\mathbb{F}}_{q^n}$ of dimension $n-k$ over ${\mathbb{F}}_q$. The existence of primitive $k$-normal elements and related properties have been studied throughout the past few years for $k>n\mathrm{/}2$. In this paper, we provide general results on the existence of primitive $k$-normal elements for the critical value $k=n\mathrm{/}2$, which have not been studied until now, except for $n=4$. Furthermore, we show the strength of this result by providing a complete characterization of the existence of primitive $3$-normal elements in ${\mathbb{F}}_{q^6}$ over ${\mathbb{F}}_q$.