Round quadratic forms (Q5896530)

A quadratic form \(\phi\) over a field K is said to be round, if \(\phi\) is hyperbolic or \(\phi\) is anisotropic and \(a\phi\) is isometric to \(\phi\) for every \(a\in K^*\) represented by \(\phi\). \textit{M. Marshall} [Math. Z. 140, 255-262 (1974; Zbl 0281.10012)] proved that every round form of dimension \(2^ v\ell\), \(\ell\) odd, is equal to \(\ell \times \psi +\rho\) in the Witt ring W(K), where \(\psi\) is a v-fold Pfister form and \(\rho\) is a torsion form. This result was applied to give a characterization of round forms over several types of fields. In the paper under review, the structure of all round forms of dimension \(2\ell (\ell odd)\) is examined. Moreover the characterization of round forms is given for linked fields and for fields with u-invariant \(\leq 4\). These results are extended to the class of fields whose Witt rings can be constructed using group extensions and products, starting with Witt rings of linked fields and of fields with \(u\leq 4\). In particular, this class contains all fields with Witt rings of elementary type.