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Translates of completely normal elements and the Morgan–Mullen conjecture
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 28 May 2026, pp. 1-21
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- Article
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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.
