Quasifields of symplectic translation planes

We show that a translation plane is symplectic if and only at least one of its associated quasifields admits a non-degenerate invariant symmetric bilinear form. As an application we prove that a proper desarguesian, Moufang or nearfield plane can never be symplectic. Moreover, we give a purely algebraic characterization of the quasifields which coordinatize symplectic translation planes.