Let \(F\) be a field of characteristic \(2\), let \(W_q(F)\), resp. \(W(F)\) denote the Witt group of nondegenerate quadratic forms, resp. the Witt ring of nondegenerate symmetric bilinear forms over \(F\), and let \(I^n(F)\) be the \(n\)-th power of the fundamental ideal of even-dimensional forms in \(W(F)\). Then \(W_q(F)\) is a \(W(F)\)-module and we put \(I^nW_q(F)=I^n(F)\otimes W_q(F)\). Let \(E/F\) be a field extension. The Witt kernel \(W_q(E/F)\) is the kernel of the natural restriction homomorphism \(W_q(F)\to W_q(E)\). Let \(L=F(\sqrt{a_1},\ldots,\sqrt{a_n})\), \(a_i\in F^*\), be a purely inseparable multiquadratic extension of \(F\), and let \(M=F(\wp^{-1}(b))\) be a separable quadratic extension of \(F\), where \(\wp^{-1}(b)\) is a root of the polynomial \(X^2+X+b\) is some algebraic closure of \(F\). The purpose of the present paper is to determine \(W_q(E/F)\) for \(E=L\) and for \(E=LM\). It is shown that \(W_q(L/F)=\sum_{i=1}^n\langle 1,a_i\rangle_b\otimes W_q(F)\), where \(\langle 1,a_i\rangle_b\) denotes the diagonal bilinear form \(x_1y_1+a_ix_2y_2\), and \(W_q(LM/F)=W_q(L/F)+W(F)\otimes [1,b]\), where \([1,b]\) denotes the quadratic form \(x^2+xy+by^2\). The proof uses differential forms and the Kato cohomology groups \(H^n_2(F)\), In fact, the authors determine first the kernels of the restriction homomorphisms \(H^n_2(F)\to H^n_2(E)\) for \(E=L, LM\) and then use K. Kato’s theorem [Invent.Math.66, 493–510 (1982; Zbl 0497.18017)] that shows that there is a natural isomorphism \(H^{n+1}_2(F)\cong I^nW_q(F)/I^{n+1}W_q(F)\).
It should be mentioned that the kernel \(W_q(L/F)\) has been computed previously by the second author [Proc.Am.Math.Soc.134, No. 9, 2481–2486 (2006; Zbl 1157.11013)], but there was mistake in the proof.