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

Abstract

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/x1dxn/xn)=(x2−x) dx1/x1dxn/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 Kato's 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 IW(F) is the maximal ideal of even-dimensional bilinear forms over F, we deduce from the above result the analogue in characteristic 2 of Knebusch's degree conjecture, i.e. InWq(F) is the set of all classes