Resolvent Components of Completely Normal Elements and the Morgan-Mullen Conjecture

Abstract

We study completely normal elements of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, that is, elements that are normal over every intermediate field, through partial Lagrange resolvents. In these coordinates, complete normality splits into independent components, in a way closely related to Hachenberger's decomposition theory. Our main tool is a structure theorem for the components; together with a finite computation, it shows that every component has weight (the sum of the probabilities that its conditions fail) at most $3/4$, uniformly in $n$. Combined with the classical character-sum method, this gives an explicit sufficient criterion for the existence of a primitive completely normal element. We verify the criterion for all $n>q$ except $70$ pairs $(q,n)$, which we settle with explicit elements found by computer. Since the case $n\le q$ is due to Garefalakis and Kapetanakis, this confirms the conjecture of Morgan and Mullen (1996) that every $\mathbb{F}_{q^n}$ contains a primitive element that is completely normal over $\mathbb{F}_q$. The same estimates give two further results. First, Hachenberger's conjectured lower bound $(q-1)^{n'}q^{(p^a-1)n'}$ for the number of completely normal elements, where $p$ is the characteristic and $n=p^an'$ with $p\nmid n'$, holds for all pairs $(q,n)$; it was known for regular pairs. Second, primitivity and complete normality are asymptotically independent: as $q^n\to\infty$, the number of primitive completely normal elements is $(1+o(1))\,φ(q^n-1)/(q^n-1)$ times the number of completely normal elements (for fixed $n$ this was known). Our arguments for $n>q$ and for both further results, including every computer-assisted step, are formalised in Lean 4 with Mathlib; the formalisation takes the case $n\le q$ and an explicit bound of Rosser and Schoenfeld as axioms and relies on Lean's compiler and interpreter for the large computations.