Maximal nonassociativity via nearfields.

Finite Fields and Their Applications 62 (2020), 101610.

We say that (x,y,z)\in Q^3 is an associative triple in a quasigroup Q(*) if (x*y)*z=x*(y*z). It is easy to show that the number of associative triples in Q is at least |Q|, and it was conjectured that quasigroups with exactly |Q| associative triples do not exist when |Q|>1. We refute this conjecture by proving the existence of quasigroups with exactly |Q| associative triples for a wide range of values |Q|. Our main tools are quadratic Dickson nearfields and the Weil bound on quadratic character sums.