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

Given a field K of characteristic 2, let W ( 2 n − 1 , K ) be the symplectic polar space defined in PG ( 2 n − 1 , K ) by a non-degenerate alternating form of V ( 2 n , K ) and Q ( 2 n , K ) be the quadric of PG ( 2 n , K ) associated to a non-singular quadratic form of Witt index n. In the literature, it is often claimed that W ( 2 n − 1 , K ) ≅ Q ( 2 n , 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.