The behavior of differential, quadratic and bilinear forms under purely inseparable field extensions

Let F be a field of characteristic p and let E/F be a purely inseparable field extension. We study the group Hpn+1(F):=coker(℘:ΩFn→ΩFn/dΩFn−1) of classes of differential forms under the extension E/F and give a system of generators of Hpn+1(E/F). In the case p=2, we use this to determine the kernel Wq(E/F) of the restriction map Wq(F)→Wq(E) between the groups of nonsingular quadratic forms over F and over E. We also deduce the corresponding result for the bilinear Witt kernel W(E′/F) of the restriction map W(F)→W(E′), where E′/F denotes a modular purely inseparable field extension.