Rund forms over real algebraic function fields in one variable

The isometry types of rund quadratic forms over an arbitrary real algebraic function field in one variable are completely determined. Using his theory of rund forms, Witt was able to simplify proofs for some of the well-known structure theorems for the Witt ring W(F) (see [6]). Thus the determination of all rund forms over a field F became a desirable goal. Using local-global methods, Hsia and Johnson were able to compute all rund forms over a global field (see [4]) and over R(t) (see [5]). In this note, using the theory of Pfister forms, we determine all rund forms over an algebraic extension field F over R(t). In particular, we show that an even dimensional rund form over F is isometric to a quadratic form of the type r(l, x)IK1, wx) for some integer r>O, where x E F and w is a sum of two squares in F. The notation in this note will follow [3]. Thus by a field F, we shall mean one whose characteristic is different from two. A quadratic form ?1 (1, xi), xi E P=F-{0} will be called an n-fold Pfister form and be notated by ((xl, , x")). The n-fold Pfister forms generate IIF as an abelian group, where IF is the ideal in the Witt ring W(F) consisting of (nonsingular) even dimensional quadratic forms. If q is a quadratic form, we shall write DF(q)=D(q) for the set of nonzero values represented by q, and GF(q)=G(q) for the group of similarity factors {x E t: (x)q q} of q. A nonsingular quadratic form q is called rund over F if either D(q)= G(q) and q is anisotropic or q is hyperbolic. Thus if q is rund, ((-x))q=O (in W(F)) for all x E D(q). It is well known that Pfister forms are rund, and we shall use this fact implicitly throughout this note. We begin by classifying all odd dimensional rund forms over an arbitrary field. LEMMA. Let F be an arbitrary field. If an odd dimensional form q over F is rund then q_(2r+ 1)(1 )for some integer r>O. The form (1) is always Received by the editors February 5, 1973. AMS (MOS) subject classfications (1970). Primary 15A63.