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

Let $F$ be a field of characteristic $p$ and let $E\mathrm{/}F$ 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 $H_p^{n+1}(E\mathrm{/}F)$. In the case $p=2$, we use this to determine the kernel $W_q(E\mathrm{/}F)$ of the restriction map $W_q(F) \to W_q(E)$betweenthegroupofnonsingularquadraticformsover$F$andover$E$\mathrm{.}WealsodeducethecorrespondingresultforthebilinearWittkernel$W(E’/F)$oftherestrictionmap$W(F) \to W(E’)$,where$E’/F$ denotes a modular purely inseparable field extension.