Round quadratic forms under algebraic extensions

Abstract

Pfister forms over fields are those anisotropic forms that remain round under any field extension. Here, round means that for any represented element x Φ 0 the isometry xφ = φ holds where φ is the form under consideration. We investigate whether a similar characterization can be given for the round forms themselves. We obtain several and theorems. Some counter- examples are given which show that a general theorem holds neither in the going-up nor in the going-down situation. Introduction. Pfister forms over fields can be characterized as those anisotropic forms that remain round under any field extension (cf. (16, p. 153)). In this paper we investigate whether a similar characteriza- tion can be given for round forms over fields. Since the structure of round forms is not known in general, one cannot expect general results. According to a theorem of Marshall (14) (see also Becker/Kόpping (3)), every round form has a decomposition φ = I x ψ + p where ψ is a Pfister form and p is torsion. This implies that any round form remains round over the Pythagorean closure of the underlying field. In this paper we shall prove the following characterization theorem for certain types of forms or fields: A form φ over F is round iff it is round over every proper quadratic extension K = F(y/w) where w is represented by φ over F. As to the going-down part of this equiva- lence the usual techniques (norm principles) allow one to prove many results. The going-up part, however, requires detailed information on the structure of round forms which is available only for certain classes of fields, for example the linked fields. Counter-examples show that in general neither implication of the equivalence is true. For extensions of odd dimension we have the well-known theorem of Springer which yields immediately that a form over F which is round over K > F ((K : F) e 2N + 3) is also round over F. In the other direction, nothing is known. We use the standard terminology as is found in (16). The fields occurring in this paper are commutative and of characteristi c Φ 2. K usually denotes a field, W{K) the corresponding Witt ring, and