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.