Given a field extension \(E/F\), the group \(W_q(E/F)\) is defined to be the kernel of the natural restriction map \(W_q(F) \rightarrow W_q(E)\). Similarly, \(_2Br(E/F)\) is the kernel of the restriction map \(_2Br(F) \rightarrow {_2Br(E)} \).
The goal of this paper is to study \(W_q(E/F)\) and \(_2Br(E/F)\) given that \(E/F\) is a simple quartic extension and \(\text{char}(F)=2\). Let \(E=F[\lambda : \lambda^4+a \lambda^3+b \lambda^2+c \lambda+d=0]\), and \(f_C(X)=X^3+b X^2+a c X+a^2 d+c^2\) be the underlying cubic resolvent. One of the following cases holds:

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The field extension is separable, in which case we may assume \(b=0\) and \(a \neq 0\).

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The field extension is inseparable but not purely inseparable, in which case \(a=c=0\) and \(b \neq 0\).

–

The field extension is purely inseparable, in which case \(a=b=c=0\).

The main result of this paper (Theorem 5.4) states that \(W_q(E/F)\) is generated as a \(W(F)\)-module by the following quadratic forms:

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\([1,g]\) for all \(g \in F\) with \(F[\wp^{-1}(g)] \subseteq E\).

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\(\langle \langle g,h]]\) for all \(h \in F\) and \(g \in F^\times\) with \(F[\sqrt{g}] \subseteq E\).

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\(\langle \langle f_C(e),\frac{d}{e^2} ]]\) for all \(e \in F^\times\) with \(f_C(e) \neq 0\).

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In Case (2), also \(\langle \langle b,d,h]]\) for all \(h \in F\).

As a result, the authors prove (Theorem 6.2) that under the same conditions, \(_2Br(E/F)\) is generated by the following quaternion algebras:

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\((h,g]\) for all \(h \in F^\times\) and \(g \in F\) with \(F[\wp^{-1}(g)] \subseteq E\).

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\((g,h]\) for all \(h \in F\) and \(g \in F^\times\) with \(F[\sqrt{g}] \subseteq E\).

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\((f_C(e),\frac{d}{e^2}]\) for all \(e \in F^\times\) with \(f_C(e) \neq 0\).