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

  • Florian Hellwig
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Abstract

We study completely normal elements of 𝔽qn over 𝔽q, 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≀ q is due to Garefalakis and Kapetanakis, this confirms the conjecture of Morgan and Mullen (1996) that every 𝔽qn contains a primitive element that is completely normal over 𝔽q. The same estimates give two further results. First, Hachenberger's conjectured lower bound (qβˆ’1)n'q(paβˆ’1)n' for the number of completely normal elements, where p is the characteristic and n=pan' with pnmid n', holds for all pairs (q,n); it was known for regular pairs. Second, primitivity and complete normality are asymptotically independent: as qnβ†’βˆž, the number of primitive completely normal elements is (1+o(1)) Ο†(qnβˆ’1)/(qnβˆ’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≀ q and an explicit bound of Rosser and Schoenfeld as axioms and relies on Lean's compiler and interpreter for the large computations.

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Cite

BibTeXRIS
BibTeX
@misc{hellwig2026resolvent,
  title = {{Resolvent Components of Completely Normal Elements and the Morgan-Mullen Conjecture}},
  author = {Florian Hellwig},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36534},
  eprint = {2610.36534},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36534},
}
RIS
TY  - UNPB
TI  - Resolvent Components of Completely Normal Elements and the Morgan-Mullen Conjecture
AU  - Florian Hellwig
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.36534
UR  - https://arxiv.org/abs/2610.36534
AB  - We study completely normal elements of 𝔽_q^n over 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≀ q is due to Garefalakis and Kapetanakis, this confirms the conjecture of Morgan and Mullen (1996) that every 𝔽_q^n contains a primitive element that is completely normal over 𝔽_q. The same estimates give two further results. First, Hachenberger's conjectured lower bound (qβˆ’1)^n'q^(p^aβˆ’1)n' for the number of completely normal elements, where p is the characteristic and n=p^an' with pnmid n', holds for all pairs (q,n); it was known for regular pairs. Second, primitivity and complete normality are asymptotically independent: as q^nβ†’βˆž, the number of primitive completely normal elements is (1+o(1)) Ο†(q^nβˆ’1)/(q^nβˆ’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≀ q and an explicit bound of Rosser and Schoenfeld as axioms and relies on Lean's compiler and interpreter for the large computations.
ER  -