Classification of Subspaces in Spaces with Definite Forms

Abstract

In the whole chapter (E, Φ) will be a positive definite hermitean space of dimension א_o over the divisionring k with involution ξ ⟼ ξ^τ. If τ ≠ 1 then it follows from Dieudonné’s lemma that k is either a quadratic extension k = k_o (γ) over an ordered field (k_o, <) with 0 > γ² ∈ k_o and (x+yγ)^τ = x-yγ for all x, y ∈ k_o: or k is a quaternion algebra $(\frac{{\alpha ,\beta }} {{{k_0}}})$ with k_o ordered, α, β < 0 and τ being the usual “conjugation”. If τ = 1, possible only when k is commutative, then ϕ is symmetric and k = k_o is ordered.