Witt rings and almost free pro-2-groups

Abstract

Let F be a field of characteristic not 2, WF the Witt ring of quadratic forms over F, IF the maximal ideal consisting of even dimensional forms, and I*F its square. Let G,(2) =Gal(F(2)/F), where F(2) is the maximal 2-extension (=quadratic closure) of F. In this note we prove a decomposition theorem for WF in the case that Z’F is torsion free and use it to obtain a new proof of a theorem of Ershov on the structure of almost free pro-a-groups [E, Theorem 41. In what follows W,F denotes the torsion subgroup of WF and WredF= WF/W, F is the reduced Witt ring. The set of orderings on F will be denoted by X, and will be given the Harrison topology, with subbasis consisting of the sets H,(a) = { < E X,1 0 <a}. F is called a SAP field if every open-and-closed (clopen) subset of X, is equal to some H,(a). If R,, R, are (abstract) Witt rings then R, x R, denotes their product in the category of Witt rings [Ml. All unexplained notations and terminology about quadratic forms and Witt rings can be found in [L] and [M]. For elementary facts about profinite groups, the reader is referred to [RI. Following Kijima and Nishi [KN] the field F is called quasi-pythugorean if Z*F is torsion free (i.e., u(F) d 2; equivalently, if the Kaplansky radical of F consists of all nonzero sums of two squares [ELl, Corollary 2.91; compare [EL2, CLASSIFICATION THEOREM 23). Examples of these fields include formally real generalized Hilbert fields [K], pseudo real closed fields [P], and fields satisfying the strong Hasse Principle H4 (every totally indefinite quadratic form of dimension 4 over F is isotropic; i.e., ii(F) < 2). A formally real field satisfies H, if and only if it is a quasi-Pythagorean SAP field [ELP]. In [Ko], Koziol gives a criterion characterizing the class of quasi-Pythagorean algebraic extensions of Q; such fields necessarily satisfy H,. The best known examples of fields satisfying H, are formally real