A trace conjecture and flag-transitive affine planes (Q5940307)

Let \(q\) be an odd prime power and consider the field extension \(F_q \subseteq F_{q^6}\) with associated trace function \(Tr\). NEWLINENEWLINENEWLINEThe authors prove that for every \(s \in F_q \setminus \{ -2,-3 \}\), there exists a \((q^2+q+1)\)-th root of unity \(u\) for which \(Tr(u)=s\). NEWLINENEWLINENEWLINEThe equations \(Tr(u)=-2\) and \(Tr(u)=-3\) have been previously studied in [\textit{R. D. Baker, J. Dover, G. L. Ebert} and \textit{K. Wantz}, Des. Codes Cryptography 21, No. 1-3, 19-39 (2000; Zbl 0970.51006)]. NEWLINENEWLINENEWLINEAs an application all odd order threedimensional flag-transitive affine planes admitting a cyclic transitive action on the line at infinity are enumerated.