Quadratic residues and character sums over fields of square order (Q1057320)

For the finite field \(F_ 2=GF(q^ 2)\), where q is an odd prime power, denote by E(k) the set of kth roots of unity in \(F_ 2\). The main objective of this paper is a proof of the following theorem: ''For \(\alpha\) \(\neq 0\) in \(F_ 2\) let \(S(\alpha)=\{\zeta +\alpha:\) \(\zeta \in E((q+1))\}\). Then S(\(\alpha)\) consists entirely of non-squares in \(F_ 2\) if and only if \(q\equiv 3 mod 4\) and \(\alpha \in E(q+1)/E((q+1))\). Further, S(\(\alpha)\) consists entirely of squares in \(F_ 2\) if and only if \(\alpha \in E((q+1))\); these squares exclude 0 if and only if \(q\equiv 1 mod 4''.\) The motivation for this theorem is a construction of M. J. Ganley for a class of weak nucleus semifields leading to a new family of projective planes of order \(q^ 4\). In his proof the author makes use of the modified Jacobsthal sum, \(\sum \chi (x^{2(q-1)}+\alpha)\), where the summation extends over all non-zero x in \(F_ 2\).