Translates of completely normal elements and the Morgan-Mullen conjecture
Theodoulos Garefalakis, Giorgos Kapetanakis
TL;DR
This work advances the Morgan–Mullen conjecture by developing new constructions of completely normal elements and proving the existence of primitive, $q^n/q$-completely normal elements in new families of finite field extensions. Central to the approach are the translate-based constructions and the notion of $(n_1,n_2)$-partially completely basic extensions, for which a main condition involving combinatorial bounds on divisor functions $W$ yields the MM property. The authors also derive an asymptotic MM result, and they implement a computational algorithm to verify the MM property for small $n_1$ (notably $n_1=2,3$) and obtain partial results for $n_1=4$, using SageMath. Collectively, these results broaden the catalog of extensions known to possess the MM property and provide a practical framework to push toward a full proof of Morgan–Mullen.
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 $\mathbb F_{q^n}^*$ 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.