On the nonexcellence of field extensions \(F(\pi)/F\)
Let \(\varphi\) be an \(n\)-fold multiplicative quadratic form over \(F\), let \(F (\varphi)\) be the function field of the quadric \(\varphi=0\). For \(n=2\) a well-known result of Arason tells us that the field extension \(F(\varphi)/F\) is excellent, i.e. for every form \(\psi\) over \(F\) the anisotropic part of \(\psi_{F(\varphi)}\) is induced from a form over \(F\). In contrast to this the author proves that for \(n\geq 3\) there exists a suitable unirational extension \(E/F\) such that \(E(\varphi)/E\) is not excellent. He constructs explicit \(2^n\)-dimensional forms \(\psi\) over \(E\) such that \((\psi_{E(\varphi)})_{an}\) is not defined over \(E\) and \(\psi\) is \(E(\varphi)\)-minimal in the sense of Hoffmann. The proof uses only results and methods from the theory of quadratic forms. In a forthcoming paper the author generalizes these results to function fields of Severi-Brauer varieties. For this he needs much deeper ingredients about unramified cohomology groups and Chow groups.
scientific article; zbMATH DE number 897861