Existence of primitive $1$-normal elements in finite fields

An element $\alpha \in {\mathbb{F}}_{q^n}$ is \emph{normal} if $$\mathcal{B}$ = {\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}}$formsabasisof$\mathbb F_{q^n}$asavectorspaceover$\mathbb F_{q}$;inthiscase,$$\mathcal{B}$$ is a normal basis of ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$. The notion of $k$-normal elements was introduced in Huczynska et al (2013). Using the same notation as before, $\alpha$ is $k$-normal if $\mathcal{B}$ spans a co-dimension $k$ subspace of ${\mathbb{F}}_{q^n}$. It can be shown that $1$-normal elements always exist in ${\mathbb{F}}_{q^n}$, and Huczynska et al (2013) show that elements that are simultaneously primitive and $1$-normal exist for $q \geq 3$andforlargeenough$n$when$\gcd(n,q) = 1$(wenotethatprimitive$1$-normalscannotexistwhen$n=2$). In this paper, we complete this theorem and show that primitive, $1$-normal elements of ${\mathbb{F}}_{q^n}$ over $\mathbb F_{q}$existforallprimepowers$q$andallintegers$n \geq 3$, thus solving Problem 6.3 from Huczynska, et al (2013).