The behavior of quadratic and differential forms under function field extensions in characteristic two

Let F be a field of characteristic 2. Let ΩnF be the F-space of absolute differential forms over F. There is a homomorphism :ΩnF→ΩnF/dΩn−1F given by (x dx1/x1 dxn/xn)=(x2−x) dx1/x1 dxn/xn mod dΩFn−1. Let Hn+1(F)=Coker( ). We study the behavior of Hn+1(F) under the function field F(φ)/F, where φ= b1,…,bn is an n-fold Pfister form and F(φ) is the function field of the quadric φ=0 over F. We show that . Using Katoʹs isomorphism of Hn+1(F) with the quotient InWq(F)/In+1Wq(F), where Wq(F) is the Witt group of quadratic forms over F and I W(F) is the maximal ideal of even-dimensional bilinear forms over F, we deduce from the above result the analogue in characteristic 2 of Knebuschʹs degree conjecture, i.e. InWq(F) is the set of all classes with deg(q) n.