Quadratic Forms Over Formally Real Fields and Pythagorean Fields

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. Contents. 0. Introduction. 1. Terminology. 2. General results. 3. Separation of orderings by quadratic forms. 4. Separation of quadratic forms by orderings. 5. Pythagorean fields. 0. Introduction. The main object of this paper is to investigate and expand the theory of quadratic forms over formally real fields. Various results on Pfister forms, Witt rings, algebraic k-groups, and Stiefel-Whitney classes are obtained. The notion of separation of sets of orderings by square classes, as well as the dual notion of separation of sets of square classes by orderings, are investigated in detail. Numerous examples are given to illustrate the general theory.In the first section, we establish the basic notations and terminology to be used throughout the paper. Following [5], we set up a topological structure on the set of all orderings of a (formally real) field, and we introduce the Strong and Weak Approximation Properties (SAP and WAP) which will be studied in detail in Sections 3, 4, and 5.The first part of Section 2 consists of preparatory results, put together there for organizational purposes. Most important is the result Theorem 2. 1 which gives several criteria for one Pfister form to be a multiple of anther. This theorem will be used freely and repeatedly throughout the paper. Next we establish a structure theorem for the torsion subgroup of I2F (I (F) (lenotes the ideal of all even dimensional forms in the Wit.t ring W(F)), and derive some sufficient conditions for a power of I(F) to be torsion-free. 1156RICHARD ELMAN AND T. Y. LAM. w characterize quadratic forms iff 3F is torsion-free. We then present somei partial calculations of ker(w), and investigate the problem of existence of quadratic forms with prescribed Stiefel-Whitney classes.Section 3 investigates the problem of separating orderings by quadratic forms. We show that given two disjoint closed sets of orderings A, B, there exists a form f C InF for some n, such that f has signature 0 in any real closure Fa (a EA) of F, while f has signature 2n in any real closur-e Ffl (,8 C B) of F. Using this 'normality property' together with a compactness argument, we obtain several alternative characterizations of WAP (Theorem 3. 3). We then show that WAP is actually equivalent to SAP (Theorem 3.5). We also relate these concepts to 'stable linkage' and show all three are equivalent. Next we define the concept of i-stability (I!+1F -21iF) for the Witt ring, and obtain some preliminary results about such Witt rings.Section 4 is parallel to Section 3, with orderings and square classes playing revers...

The reduced Witt ring of a formally real field

“…F is an exact field precisely when equality holds in one (and hence in both) of (8) or (9). If F is exact, then Wted(F) is determined up to ring isomorphism by the following basic invariants of F: the spaces 6(F) and 911(F), the groups Aa and Agr(o,T E 911(F)), and the maps wOT and (P,T) (o,t E 911(F) and P, T E 0O).…”

“…We introduce certain parity and continuity conditions which elements of this subdirect product must satisfy; we call those fields "exact" for which these conditions characterize elements of the subdirect product. Some classes of exact fields are presented in §4; particular attention is paid to SAP fields [9, Definition 1.4], superpythagorean fields [9,Definition 4.4], rational function fields, and Henselian valued fields. An apparently weaker condition than exactness, "near exactness", is studied in §5; for these fields one can again describe the reduced Witt ring as a subdirect product of the Witt rings of the ultracompletions, but only by involving less simple invariants of the field.…”

A note on quadratic forms over Pythagorean fields

“…Hence the implication is obvious in this case. Now assume F is formally real and let Tr* denote Scharlau's transfer map relative to the F-linear trace map Tr L/F (which associates to each quadratic form q over L the F-quadratic form Tr L/F°q ) [4,Chapter 7,§ 1,6], [3, §5]. Then for any anisotropic form q over F, there is an isometry [4,Theorem 3.3], [L : F] q is anisotropic over F. Therefore Tr*(q L ) is anisotropic over F so that, in particular, q L is anisotrpic over L.…”

“…The field F satisfies the Strong Approximation Property (SAP) if given any two disjoint closed subsets ί/, V of X there is an element a in F which is positive at the orderings in U and negative at the orderings in V (cf. [1,Definition 1.4], [3,Corollary 3.21]). is injective then < extends to L so an equation a x x?+ + a n x n 2 = 0 with each x t in L is impossible.…”

Levels of division algebras

“…We let D F (q) denote the nonzero elements of F represented by a quadratic form q over F. The topological space of orderings of a field F is denoted X F . Basic properties of X F and basic results on SAP fields can be found in [3] and [11]. …”