Translates of completely normal elements and the Morgan-Mullen conjecture

Abstract

Denote by ${\mathbb{F}}_q$ the finite field of order $q$ and by ${\mathbb{F}}_{q^n}$ its extension of degree $n$. Some $a\in {\mathbb{F}}_{q^n}$ is called primitive if it generates the multiplicative group ${\mathbb{F}}_{q^n}^{\ast }$ and it is called $q^n/q$-normal if its ${\mathbb{F}}_q$-conjugates form an ${\mathbb{F}}_q$-basis of ${\mathbb{F}}_{q^n}$ if the latter is viewed as an ${\mathbb{F}}_q$-vector space. Furthermore, some $a\in {\mathbb{F}}_{q^n}$ is called $q^n/q$-completely normal if it is $q^n/q^d$-normal for all $d\mid n$. In this work we prove a new construction of sets of completely normal elements and, we establish, under conditions, the existence of elements that are simultaneously primitive and $q^n/q$-completely normal, covering some yet unresolved cases of a 30-year-old conjecture by Morgan and Mullen.