Semifields in prime dimensions and counterexamples to Kaplansky's conjecture
Abstract
In 1975, Kaplansky conjectured that every five-dimensional division algebra over a sufficiently large finite field is a field or a twisted field. We disprove this conjecture. For every prime power q=pe≡1pmod3 and every n≥5 with gcd(n,6)=1, we construct semifields of order qn. For fixed q,n, the family represents φ(n) isotopy classes if p≡1pmod3, and φ(n)/2 if p≡2pmod3, where φ is Euler's totient function. Using new isotopy invariants and the structural properties of our construction, we prove that none of these semifields is isotopic to a finite field or an Albert's generalized twisted field. In particular, the five-dimensional specialization gives infinitely many pairwise nonisotopic counterexamples to Kaplansky's conjecture over arbitrarily large finite fields. More generally, for each prime dimension n≥5, the examples occur over arbitrarily large fields in every characteristic other than three, contradicting the classification asserted by Menichetti in 1996, whose proof contains gaps.
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BibTeX
@misc{nagy2026semifields,
title = {{Semifields in prime dimensions and counterexamples to Kaplansky's conjecture}},
author = {Gábor P. Nagy and Yue Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32651},
eprint = {2609.32651},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32651},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Semifields in prime dimensions and counterexamples to Kaplansky's conjecture
AU - Gábor P. Nagy
AU - Yue Zhou
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32651
UR - https://arxiv.org/abs/2609.32651
AB - In 1975, Kaplansky conjectured that every five-dimensional division algebra over a sufficiently large finite field is a field or a twisted field. We disprove this conjecture. For every prime power q=p^e≡1pmod3 and every n≥5 with gcd(n,6)=1, we construct semifields of order q^n. For fixed q,n, the family represents φ(n) isotopy classes if p≡1pmod3, and φ(n)/2 if p≡2pmod3, where φ is Euler's totient function. Using new isotopy invariants and the structural properties of our construction, we prove that none of these semifields is isotopic to a finite field or an Albert's generalized twisted field. In particular, the five-dimensional specialization gives infinitely many pairwise nonisotopic counterexamples to Kaplansky's conjecture over arbitrarily large finite fields. More generally, for each prime dimension n≥5, the examples occur over arbitrarily large fields in every characteristic other than three, contradicting the classification asserted by Menichetti in 1996, whose proof contains gaps.
ER -