Function field extensions and annihilators of differential forms in characteristic p and bilinear forms in characteristic 2

Abstract

Let F be a field of characteristic p > 0 and let Ω n (F ) be the F -vector space of n-differential forms over F . In this work we will study the behaviour of Ω n (F ) under iterated function field extensions of p-forms. We will use results from a previous work to rewrite the kernel of the restriction map Ω n (F ) → Ω n (F (ϕ 1 , . . . , ϕ r )) to a set of differential forms annihilated by specific forms given by the norm fields of ϕ 1 , . . . , ϕ r .W (F (π)/F ) = y 1 , . . . , y r | y 1 , . . . , y r ∈ F 2 (a 1 , . . . , a r ) * and W q (F (π)/F ) = π ⊗ W q (F ).