Translates of completely normal elements and the Morgan–Mullen conjecture

Denote by F q $\mathbb F_q$ double struck upper F Subscript q the finite field of order q and by F q n $\mathbb F_{q^n}$ double struck upper F Subscript q Sub Superscript n its extension of degree n . Some a ∈ F q n $a\in \mathbb F_{q^n}$ a element of double struck upper F Subscript q Sub Superscript n is called primitive if it generates the multiplicative group F q n ∗ $\mathbb F_{q^n}^*$ double struck upper F Subscript q Sub Superscript n Superscript asterisk , and it is called q n / q $q^n/q$ q Superscript n Baseline divided by q -normal if its F q $\mathbb F_q$ double struck upper F Subscript q -conjugates form an F q $\mathbb F_q$ double struck upper F Subscript q -basis of F q n $\mathbb F_{q^n}$ double struck upper F Subscript q Sub Superscript n if the latter is viewed as an F q $\mathbb F_q$ double struck upper F Subscript q -vector space. Furthermore, some a ∈ F q n $a\in \mathbb F_{q^n}$ a element of double struck upper F Subscript q Sub Superscript n is called q n / q $q^n/q$ q Superscript n Baseline divided by q -completely normal if it is q n / q d $q^n/q^d$ q Superscript n Baseline divided by q Superscript d -normal for all d ∣ n $d\mid n$ d vertical bar 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 $q^n/q$ q Superscript n Baseline divided by q -completely normal, covering some yet unresolved cases of a 30-year-old conjecture by Morgan and Mullen.