Round quadratic forms (Q5917689)

The author has been motivated by the wish to formulate the results of \textit{B. Alpers} [J. Algebra 137, 44-55 (1991; Zbl 0727.11017)]\ on round quadratic forms for quadratic forms instead of for elements of the Witt ring. The goal of this paper is finding a classification of round quadratic forms. Here the \(u\)-invariant plays a role as do some new invariants that the author defines. Let \(F\) be a field of characteristic \(\neq 2\). A nonsingular quadratic form \(q: F^{2n}\to F\) is said to be round iff \(D(q)= G(q)\), where \(D(q)= q(F^{2n}) \setminus \{0\}\) and \(G(q)= [a\in F^*\mid q\cong\langle a\rangle q\}\). The author lays the foundations to this paper in part 2 devoted to theorems. These he applies in part 3 to characterize round forms under various conditions. Finally he studies special cases, in particular the dimension 4 case, linked fields and fields \(F\) with \(u(F)=4\).