ROUND QUADRATIC-FORMS UNDER ALGEBRAIC EXTENSIONS

Pfister forms over fields are those anisotropic forms that remain round under any field extension. Here, round means that for any represented element x not-equal 0 the isometry x phi congruent-to phi holds where phi is the form under consideration. We investigate whether a similar characterization can be given for the round forms themselves. We obtain several "going-up" and "going-down" theorems. Some counter-examples are given which show that a general theorem holds neither in the going-up nor in the going-down situation.