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

Given a field K of characteristic 2 and an integer n ≥ 2, let W (2n − 1, K) be the symplectic polar space defined in PG(2n − 1, K) by a nondegenerate alternating form of V (2n, K) and let 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 otherwise. In this paper we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that W (2n − 1, K) is indeed isomorphic to a non-singular quadric Q, but when K is nonperfect the nucleus of Q has vector dimension greater than 1. So, in this case, Q(2n, K) is a proper subgeometry of W (2n − 1, K). We show that, in spite of this fact, W (2n− 1, K) can be embedded in Q(2n, K) as a subgeometry and that this embedding induces a full embedding of the dual DW (2n−1, K) of W (2n−1, K) into the dual DQ(2n, K) of Q(2n, K).