  Let $F$ be a field of characteristic $p$ and let $E/F$ be a purely
inseparable field extension. We study the group $H_p^{n+1}(F)$ of classes of
differential forms under the restriction map $H_p^{n+1}(F)\to H_p^{n+1}(E)$ and
give a system of generators of the kernel $H_p^{n+1}(E/F)$. In the case $p=2$,
we use this to determine the kernel $W_q(E/F)$ of the restriction map $W_q(F)
\to W_q(E)$ between the group 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 $W(F) \to W(E')$, where $E'/F$ denotes a
modular purely inseparable field extension.
