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 $q=p^e\equiv 1(\mathrm{mod}3)$ and every $n\ge 5$ with $\gcd (n,6)=1$, we construct semifields of order $q^n$. For fixed $q,n$, the family represents $\phi (n)$ isotopy classes if $p\equiv 1(\mathrm{mod}3)$, and $\phi (n)/2$ if $p\equiv 2(\mathrm{mod}3)$, where $\phi$ 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 $n\ge 5$, 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.