Semifields in prime dimensions and counterexamples to Kaplansky's conjecture
Gábor P. Nagy, Yue Zhou
math.CO
Sep 26, 2026 · v1
TL;DR
The paper's results were formalized by Harmonic's Aristotle and machine-checked in Lean 4, with the formalization cited as a reference.
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 $\varphi(n)$ isotopy classes if $p\equiv1\pmod3$, and $\varphi(n)/2$ if $p\equiv2\pmod3$, where $\varphi$ 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.
Problem
Kaplansky conjectured in 1975 that every five-dimensional division algebra over a sufficiently large finite field is a field or a twisted field. Menichetti claimed a related classification for prime dimensions in 1996, but his proof contains gaps.
Approach
For prime powers q ≡ 1 (mod 3) and n ≥ 5 with gcd(n,6)=1, the authors build explicit presemifields on F_{q^n} from a bilinear multiplication defined with Frobenius twists and a cube root of -1. They compute the split left determinant and the nuclei. They then introduce new isotopy invariants, component splitting fields and determinant-vertex ranks, to separate the family from finite fields and Albert's generalized twisted fields. The results were developed with ChatGPT assistance, then formalized by Aristotle and verified in Lean 4.
Results
The family gives φ(n) isotopy classes, or φ(n)/2 when p ≡ 2 (mod 3). None of them is isotopic to a field or a generalized twisted field. This yields infinitely many nonisotopic counterexamples to Kaplansky's conjecture in dimension five and contradicts Menichetti's claimed classification.
