Witt kernels of quadratic forms for multiquadratic extensions in characteristic 2

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 $2$.Comment: 9 page

Witt kernels and Brauer kernels for quartic extensions in characteristic two

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