A Trace Conjecture and Flag-Transitive Affine Planes

For any odd prime power q, all (q2−q+1)th roots of unity clearly lie in the extension field Fq6 of the Galois field Fq of q elements. It is easily shown that none of these roots of unity have trace −2, and the only such roots of trace −3 must be primitive cube roots of unity which do not belong to Fq. Here the trace is taken from Fq6 to Fq. Computer based searching verified that indeed −2 and possibly −3 were the only values omitted from the traces of these roots of unity for all odd q⩽200. In this paper we show that this fact holds for all odd prime powers q. As an application, all odd order three-dimensional flag-transitive affine planes admitting a cyclic transitive action on the line at infinity are enumerated.