Quadratic forms, division algebras and inseparable multiquadratic extensions
For a field \(k\) of characteristic 2 let \(Wq(k)\) be the Witt group of quadratic forms over \(k\). Let \(K\) be an extension field of \(k\) and let \(\iota^*_q: Wq(k)\to Wq(K)\) be the natural morphism. The authors consider the problem of determining the kernel of \(\iota^*_q\) in case of purely inseparable extensions \(K/k\). The case of separable extensions (of characteristic 2) has been investigated earlier, as well as the case of inseparable quadratic extensions [cf. \textit{R. Baeza}, Quadratic forms over semilocal rings, Lect. Notes Math. 655 (Springer Verlag 1978; Zbl 0382.10014)]. When \(K= k( \sqrt{a_1}, \dots, \sqrt{a_n})\) the authors show that \[ \ker \iota^*_q= \langle 1, a_1\rangle Wq(k)+ \cdots+ \langle 1,a_n \rangle Wq(k) \] holds for \(n=2\), and also for all \(n\geq 1\) provided that \(I^2 Wq(k)=0\). A basic observation is that an inseparable quadratic extension \(K/k\) is excellent (in the sense that the anisotropic part of every quadratic form over \(K\) is defined over \(k\)). The paper ends with two results on the structure of division \(k\)-algebras of exponent 2 which contain largely purely inseparable extensions of the ground field.
scientific article; zbMATH DE number 817249
scientific article