Quadratic residues and character sums over fields of square order

The modified Jacobsthal sum Σ_{x ∈ GF^?(q²)}χ(x^{2(q ? 1)} + α) is estimated yielding a classification of the α in GF(q²) for which ξ + α is always a non-square (or a non-zero square) for all $\text{1}\text{2}\text{(q + 1)}\text{th}$ roots of unity ξ. The motivation is that this provides every member of a new family of projective planes constructed by M. J. Ganley.