Partitions of three-dimensional Pappian projective spaces by ovoidal quadrics (Q1128191)

Consider a commutative field \(K\) and a separable extension \(V/K\) of degree \(4\) such that there is exactly one intermediate extension \(E/K\). If \(\rho\) denotes the involutory automorphism of \(V/E\), then the condition \(xx^\rho \in K\) defines an ovoidal quadric \(C\), and \(\Pi = \{Ca \mid a \in V^\times\}\) is a partition of the \(3\)- dimensional projective space \(\text{ P}V\). This is true regardless of the characteristic \(\chi\) of \(K\), but the proofs distinguish between \(\chi \neq 2\) and \(\chi = 2\). For finite fields the result is due to \textit{G. L. Ebert} [Can. J. Math. 37, 1163-1175 (1985; Zbl 0577.51001)].