Hyperplanes of \(DW(5,{\mathbb{K}})\) with \({\mathbb{K}}\) a perfect field of characteristic 2 (Q968242)

Let \({\mathbb{K}}\) be a perfect field of characteristic 2. The author classifies all hyperplanes of the symplectic dual polar space \(DW(5,{\mathbb{K}})\) that arise from its Grassmann embedding. Also, the author shows 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 \(\{\lambda\in {\mathbb{K}}\mid X^{2}+\lambda X+1 \text{ is irreducible in }{\mathbb{K}}[X]\} : (\lambda _{1},\lambda _{2})\in R\) whenever there exists an automorphism \(\sigma \) of \({\mathbb{K}}\) and an \(a\in {\mathbb{K}}\) such that \((\lambda^{\sigma}_2)^{-1}=\lambda_1^{-1} +a^{2}+a\).