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Spaces of triangularizable matrices (III): Perfect non-quadratically closed fields with characteristic 2

  • Clément de Seguins Pazzis
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Abstract

Given a field 𝔽 and an integer n ≥ 2, denote by tn(𝔽) the greatest possible dimension for a vector space of n-by-n matrices over 𝔽 in which every element is triangularizable. It was recently proved that tn(𝔽)=(n(n+1))/2 if and only if 𝔽 is not quadratically closed, with the possible exception of finite fields with characteristic $2$ and less than n−1 elements. In this article, we prove that the equality tn(𝔽)=(n(n+1))/2 holds for all perfect non-quadratically closed fields with characteristic $2$ -- with the possible exception of fields with cardinality $2$ -- and for these fields we obtain a key result for a future analysis of the spaces that have the critical dimension tn(𝔽).

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Cite

BibTeXRIS
BibTeX
@misc{pazzis2026spaces,
  title = {{Spaces of triangularizable matrices (III): Perfect non-quadratically closed fields with characteristic 2}},
  author = {Clément de Seguins Pazzis},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.28863},
  eprint = {2608.28863},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28863},
}
RIS
TY  - UNPB
TI  - Spaces of triangularizable matrices (III): Perfect non-quadratically closed fields with characteristic 2
AU  - Clément de Seguins Pazzis
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2608.28863
UR  - https://arxiv.org/abs/2608.28863
AB  - Given a field 𝔽 and an integer n ≥ 2, denote by t_n(𝔽) the greatest possible dimension for a vector space of n-by-n matrices over 𝔽 in which every element is triangularizable. It was recently proved that t_n(𝔽)=(n(n+1))/2 if and only if 𝔽 is not quadratically closed, with the possible exception of finite fields with characteristic $2$ and less than n−1 elements. In this article, we prove that the equality t_n(𝔽)=(n(n+1))/2 holds for all perfect non-quadratically closed fields with characteristic $2$ -- with the possible exception of fields with cardinality $2$ -- and for these fields we obtain a key result for a future analysis of the spaces that have the critical dimension t_n(𝔽).
ER  -