Hyperplanes of DW ( 5 , K ) with K a perfect field of characteristic 2

Let K be a perfect field of characteristic 2. In this paper, we classify all hyperplanes of the symplectic dual polar space DW(5,K) that arise from its Grassmann embedding. We show that the number of isomorphism classes of such hyperplanes is equal to 5+N , where N is the number of equivalence classes of the following equivalence relation R on the set {λ ∈ K |X2 + λX + 1 is irreducible in K[X]}: (λ1, λ2) ∈ R whenever there exists an automorphism σ of K and an a ∈K such that (λσ2 ) −1 = λ−1 1 + a2 + a.