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

TL;DR

This paper investigates the behavior of differential forms over fields of characteristic p under iterated function field extensions, linking it to bilinear forms in characteristic 2, and provides new insights into their kernels and annihilators.

Contribution

It introduces a novel description of the kernel of restriction maps of differential forms using norm fields, and connects these results to bilinear forms in characteristic 2.

Findings

Kernel of restriction maps characterized by norm fields

Differential forms annihilated by specific forms identified

Results translated to bilinear forms in characteristic 2

Abstract

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 $2$.