Annihilators of differential forms over over fields of characteristic p

Let $F$ be a field of characteristic $p$ and let $\Omega^n(F)$ be the $F$-vector space of $n$-differential forms. In this work, we will study the annihilator of differential forms, give specific descriptions for special cases and show a connection between these annihilators and the kernels of the restriction map $\Omega^n(F) \to \Omega^n(E)$ for purely inseparable field extensions $E/F$.