Round quadratic forms under algebraic extensions (Q1119678)

Pfister forms over fields are those anisotropic forms that remain round under any field extension. Here, round means that for any non-zero represented element x the isometry \(q\cong xq\) holds where q is the form under consideration. We investigate whether a similar characterization can be given for the round forms themselves. We get several ``going-up''- and ``going-down''-theorems. Some counter-examples are given which show that neither in the ``going-up'' nor in the ``going-down'' situation a general theorem holds.