Witt kernels of quadratic forms for multiquadratic extensions in characteristic 2

TL;DR

This paper proves that purely inseparable extensions of exponent 1 in characteristic 2 are excellent for quadratic forms, enabling the extension of known results on Witt kernels for multiquadratic extensions.

Contribution

It establishes the excellence of such extensions for quadratic forms and extends the computation of Witt kernels for multiquadratic extensions in characteristic 2.

Findings

Extension is excellent for quadratic forms in characteristic 2.

Extended results on generators of Witt kernels for multiquadratic extensions.

Recovered and generalized previous results by Aravire and Laghribi.

Abstract

Let $F$ be a field of characteristic $2$ and let $K/F$ be a purely inseparable extension of exponent $1$. We show that the extension is excellent for quadratic forms. Using the excellence we recover and extend results by Aravire and Laghribi who computed generators for the kernel $W_q(K/F)$ of the natural restriction map $W_q(F)\to W_q(K)$ between the Witt groups of quadratic forms of $F$ and $K$, respectively, where $K/F$ is a finite multiquadratic extension of separability degree at most $2$.