Function field extensions and annihilators of differential forms in characteristic p and bilinear forms in characteristic 2

Let $F$ be a field of characteristic $p>0$ and let $\Omega^n(F)$ be the $F$-vector space of $n$-differential forms over $F$. In this work we will study the behaviour of $\Omega^n(F)$ under iterated function field extensions of $p$-forms. We will use results from a previous work to rewrite the kernel of the restriction map $\Omega^n(F) \to \Omega^n(F(\varphi_1,\ldots,\varphi_r))$ to a set of differential forms annihilated by specific forms given by the norm fields of $\varphi_1,\ldots,\varphi_r$. We will close this work by translating the new resluts to the theory of bilinear forms over fields of characteristic $2$.