A trace conjecture and flag-transitive affine planes

For any odd prime power q. all (q(2)-q + 1)th roots of unity clearly lie in the extension field F-q(6) of the Galois field F-q of q elements. It is ea sily shown that none of these roots of unity have trace -2. and the only su ch roots of trace -3 must be primitive cube roots of unity which do not bel ony to F-q. Here the trace is taken from F-q(6) to F-q. Computer based sear ching verified that indeed -2 and possibly -3 were the only values omitted from the traces of these roots of unity for all Odd q less than or equal to 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 en umerated. (C) 2001 Academic Press.