On $k$-normal elements over finite fields

The so called $k$-normal elements appear in the literature as a generalization of normal elements over finite fields. Recently, questions concerning the construction of $k$-normal elements and the existence of $k$-normal elements that are also primitive have attracted attention from many authors. In this paper we give alternative constructions of $k$-normal elements and, in particular, we obtain a sieve inequality for the existence of primitive, $k$-normal elements. As an application, we show the existence of primitive $k$-normal elements for a significant proportion of $k$'s in many field extensions. In particular, we prove that there exist primitive $k$-normals in $\mathbb F_{q^n}$ over $\mathbb F_q$ in the case when $k$ lies in the interval $[1, n/4]$, $n$ has a special property and $q, n\ge 420$.