This work demonstrates the existence of elements that are primitive and normal in finite fields, implying potential resolutions for the Morgan-Mullen conjecture.
Denote by Fq the finite field of order q and by Fqⁿ its extension of degree n. Some a∈ Fqⁿ is called primitive if it generates the multiplicative group Fqⁿ^* and it is called qⁿ/q-normal if its Fq-conjugates form an Fq-basis of Fqⁿ if the latter is viewed as an Fq-vector space. Furthermore, some a∈ Fqⁿ 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.
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Garefalakis et al. (2025) studied this question.