Let $F$ be a field of characteristic $2$ and let $E/F$ be a field extensionof 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, Mammoneand 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 ofnondegenerate symmetric bilinear forms over $F$ and over $E$ from earlierresults by the first author. As application, we describe the $2$-torsion partof the Brauer kernel for such extensions.
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机译：假设$ F $是特征$ 2 $的字段，而$ E / F $是度数$ 4 $的字段扩展。我们确定限制图$ W_qF \到W_qE $之间的限制图$ W_q（E / F）$的内核$ W_qE $之间的非退化二次形式超过$ F $和超过$ E $，完成了Ahmad，Baeza，Mammoneand的更早的部分结果莫雷西我们还从先前的结果中，首先减去了在$ F $和$ E $之上的非简并对称双线性形式的Witt环之间的限制图$ WF \的Witt核$ W（E / F）$的对应结果到WE $作者。作为应用程序，我们描述了Brauer内核的$ 2 $ -torsion部分用于此类扩展。
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