Denote by Fq the finite field of order q and by Fₐ䂞 its extension of degree n. Some a Fₐ䂞 is called primitive if it generates the multiplicative group Fₐ䂞^* and it is called qⁿ/q-normal if its Fq-conjugates form an Fq-basis of Fₐ䂞 if the latter is viewed as an Fq-vector space. Furthermore, some a Fₐ䂞 is called qⁿ/q-completely normal if it is qⁿ/qᵈ-normal for all d 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 qⁿ/q-completely normal, covering some yet unresolved cases of a 30-year-old conjecture by Morgan and Mullen.
Garefalakis et al. (Sat,) studied this question.