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Gauss Genus Theory in Characteristic 2

  • Qiyu Zhang
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Abstract

We extend Gauss composition and Gauss genus theory over Z to 𝔽2n[T], a polynomial ring over a finite field 𝔽2n of characteristic 2. We find new invariants of binary quadratic forms over 𝔽2n[T] by using Arf invariant and introduce new definitions of proper equivalence and direct composition, and prove that the direct composition makes the set of proper equivalence classes of binary quadratic forms with the same invariants into a finite Abelian group, which is isomorphic to a Picard group of a corresponding extension ring of 𝔽2n[T]. Building on this, we develop genus theory in characteristic 2 and prove that the kernel of the generalized Gauss's map is the subgroup of all squares in the class group.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026gauss,
  title = {{Gauss Genus Theory in Characteristic 2}},
  author = {Qiyu Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12789},
  eprint = {2609.12789},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12789},
}
RIS
TY  - UNPB
TI  - Gauss Genus Theory in Characteristic 2
AU  - Qiyu Zhang
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12789
UR  - https://arxiv.org/abs/2609.12789
AB  - We extend Gauss composition and Gauss genus theory over Z to 𝔽_2^n[T], a polynomial ring over a finite field 𝔽_2^n of characteristic 2. We find new invariants of binary quadratic forms over 𝔽_2^n[T] by using Arf invariant and introduce new definitions of proper equivalence and direct composition, and prove that the direct composition makes the set of proper equivalence classes of binary quadratic forms with the same invariants into a finite Abelian group, which is isomorphic to a Picard group of a corresponding extension ring of 𝔽_2^n[T]. Building on this, we develop genus theory in characteristic 2 and prove that the kernel of the generalized Gauss's map is the subgroup of all squares in the class group.
ER  -