A note on symplectic polar spaces over non-perfect fields of characteristic 2

Given a field K of characteristic 2, let W(2n−1,K) be the symplectic polar space defined in PG(2n−1,K) by a non-degenerate alternating form of V(2n,K) and Q(2n,K) be the quadric of PG(2n,K) associated to a non-singular quadratic form of Witt index n. In the literature, it is often claimed that W(2n−1,K)≅Q(2n,K). This is true when K is perfect, but false if otherwise. In this paper we modify the previous claim, so to obtain a statement that is correct for any field of characteristic 2.