Lower bounds for essential dimensions via orthogonal representations

Let us first recall what is the essential dimension of a functor, cf. [BR 97] and [R 00]. Let k be a field, and let F be a functor from the category of field extensions of k into the category of sets. Let F/k be an extension and let ξ be an element of F(F ). If E is a field with k ⊂ E ⊂ F we say that ξ comes from E if it belongs to the image of F(E) → F(F ). The essential dimension ed (ξ) of ξ is the minimum of the transcendence degrees E/k, for all E with k ⊂ E ⊂ F such that ξ comes from E. One has ed(ξ) ≤ tr. deg. F . If there is equality, we say that ξ is incompressible. The essential dimension ed (F) of F is ed (F) = max {ed (ξ)}, the maximum being taken over all pairs (F, ξ) with k ⊂ F and ξ ∈ F(F ).