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A note on generalized distributive sets of a finite Dickson nearfield

  • K. S. Enoch Lee
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Abstract

Let 𝔽qn be a finite field where q=pl (p is a prime) and a Dickson nearfield R associated to the field. In this note we investigate the basic properties of the generalized distributive set D(a,b) where a,b∈ R. The set is well-known when n=1, 2. For n>2, we characterize when D(a,b) is a subfield of 𝔽qn. Furthermore we find the unique largest subfield of 𝔽qn contained in D(a,b). We also provide a condition to guarantee that D(a,b) is not a subfield of 𝔽qn. We end the note by posing the question of when D(a,b), as a space over 𝔽p, has a primitive normal basis.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026note,
  title = {{A note on generalized distributive sets of a finite Dickson nearfield}},
  author = {K. S. Enoch Lee},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10228},
  eprint = {2609.10228},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10228},
}
RIS
TY  - UNPB
TI  - A note on generalized distributive sets of a finite Dickson nearfield
AU  - K. S. Enoch Lee
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10228
UR  - https://arxiv.org/abs/2609.10228
AB  - Let 𝔽_q^n be a finite field where q=p^l (p is a prime) and a Dickson nearfield R associated to the field. In this note we investigate the basic properties of the generalized distributive set D(a,b) where a,b∈ R. The set is well-known when n=1, 2. For n>2, we characterize when D(a,b) is a subfield of 𝔽_q^n. Furthermore we find the unique largest subfield of 𝔽_q^n contained in D(a,b). We also provide a condition to guarantee that D(a,b) is not a subfield of 𝔽_q^n. We end the note by posing the question of when D(a,b), as a space over 𝔽_p, has a primitive normal basis.
ER  -