Witt kernels and Brauer kernels for quartic extensions in characteristic two

TL;DR

This paper fully characterizes the Witt kernel for quadratic forms and the Brauer kernel in quartic extensions over fields of characteristic two, extending previous partial results and providing new insights into the structure of these algebraic objects.

Contribution

It completes the determination of Witt kernels for quartic extensions in characteristic two and derives the 2-torsion part of the Brauer kernel, advancing understanding of quadratic form and Brauer group relations.

Findings

Complete description of the Witt kernel W_q(E/F) for quartic extensions in characteristic two.

Derivation of the Witt kernel W(E/F) for symmetric bilinear forms from earlier results.

Description of the 2-torsion part of the Brauer kernel for such extensions.

Abstract

Let $F$ be a field of characteristic $2$ and let $E/F$ be a field extension of degree $4$. We determine the kernel $W_q(E/F)$ of the restriction map $W_qF\to W_qE$ between the Witt groups of nondegenerate quadratic forms over $F$ and over $E$, completing earlier partial results by Ahmad, Baeza, Mammone and Moresi. We also deduct the corresponding result for the Witt kernel $W(E/F)$ of the restriction map $WF\to WE$ between the Witt rings of nondegenerate symmetric bilinear forms over $F$ and over $E$ from earlier results by the first author. As application, we describe the $2$-torsion part of the Brauer kernel for such extensions.