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 ${\mathrm{\Omega }}^n(F)$ be the $F$-vector space of $n$-differential forms over $F$. In this work we will study the behaviour of ${\mathrm{\Omega }}^n(F)$ under iterated function field extensions of $p$-forms. We will use results from a previous work to rewrite the kernel of the restriction map ${\mathrm{\Omega }}^n(F)\to {\mathrm{\Omega }}^n(F({\phi }_1,\dots ,{\phi }_r))$ to a set of differential forms annihilated by specific forms given by the norm fields of ${\phi }_1,\dots ,{\phi }_r$. We will close this work by translating the new resluts to the theory of bilinear forms over fields of characteristic $2$.