About r-primitive and k-normal elements in finite fields

Abstract

In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of k-normal elements: an element $\alpha \in {\mathbb {F}}_{q^n}$ is k-normal over ${\mathbb {F}}_q$ if the greatest common divisor of the polynomials $g_{\alpha }(x)= \alpha x^{n-1}+\alpha ^qx^{n-2}+\cdots +\alpha ^{q^{n-2}}x+\alpha ^{q^{n-1}}$ and $x^n-1$ in ${\mathbb {F}}_{q^n}[x]$ has degree k, generalizing the concept of normal elements (normal in the usual sense is 0-normal). In this paper we discuss the existence of r-primitive k-normal elements in ${\mathbb {F}}_{q^n}$ over ${\mathbb {F}}_{q}$, where an element $\alpha \in {\mathbb {F}}_{q^n}^*$ is r-primitive if its multiplicative order is $\frac{q^n-1}{r}$. We provide many general results about the existence of this class of elements and we work a numerical example over finite fields of characteristic 11.