Year: 2026
Type: preprint
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 $q=p^e\equiv1\pmod3$ and every $n\ge5$ with $\gcd(n,6)=1$, we construct semifields of order $q^n$. For fixed $q,n$, the family represents $φ(n)$ isotopy classes if $p\equiv1\pmod3$, and $φ(n)/2$ if $p\equiv2\pmod3$, 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 $n\ge5$, 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.
Source: arXiv (Cornell University)
Authors: Nagy, Gábor P., Zhou, Yue
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G3293424821
OpenAlex ID: W7214791339