{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36534",
  "author_total": 1,
  "authors": [
    {
      "name": "Florian Hellwig"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5347v50r6dvs1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36534",
  "fulltext_url": null,
  "id": "5347v50r6dvs1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36534",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7224103723",
  "pdf_url": "https://arxiv.org/pdf/2610.36534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5347v50r6dvs1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Resolvent Components of Completely Normal Elements and the Morgan-Mullen Conjecture",
  "title_text": "Resolvent Components of Completely Normal Elements and the Morgan-Mullen Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/5347v50r6dvs1",
  "venue": "arXiv",
  "versions": []
}