Reduced Witt Rings Research Articles

Isomorphism criteria for Witt rings of real fields

We prove isomorphism criteria for Witt rings and reduced Witt rings of certain types of real fields. Refined criteria are obtained under the additional assumption that the field be SAP. This leads to a generalization of a result by Koprowski on Witt equivalence of function fields of transcendence degree 1 over a real closed field. Isomorphism criteria are also obtained for Witt rings of hermitian forms over a quadratic extension of a real base field and for Witt groups of hermitian forms over a quaternion algebra with a real field as center. All these criteria are expressed in terms of properties involving topological subspaces of the space of orderings of the base field.

Symbol length and stability index

We show that a Pythagorean field (more generally, a reduced abstract Witt ring) has finite stability index if and only if it has finite 2-symbol length. We give explicit bounds for the two invariants in terms of one another. To approach the question whether those bounds are optimal we consider examples of Pythagorean fields.

Torsion-Free Modules over Reduced Witt Rings

We compute the genus class group of a torsion-free module over a reduced Witt ring of finite stability index. This is applied to modules locally isomorphic to odd degree extensions of formally real fields.

Formally real fields from a Galois theoretic perspective

To each field F of characteristic not 2, one can associate a certain Galois group G F , the so-called W-group of F, which carries essentially the same information as the Witt ring W( F) of equivalence classes of anisotropic quadratic forms over F. There is a close connection between (nontrivial) involutions in G F and orderings on F. The purpose of this paper is to investigate how the lattice of orderings and preorderings on F is determined by G F , and to provide a Galois-theoretic version of reduced Witt rings.

Valuations and higher level orders in commutative rings

Valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields, see, for example [BR]. In [MW], the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting. While it seems desirable to extend these results further, the techniques used for rings with many units will not work for general commutative rings. At the same time, there is a general theory of valuations in commutative rings (see [LM], [M], and [G]), which in [Ma] was used to study orders and the reduced theory of quadratic forms over general commutative rings. Thus it seems natural to ask if the connections between valuations and higher level orders in fields exist in commutative rings. In this paper we use valuation theory to study the space of orders and the reduced Witt ring relative to a higher level preorder in a commutative ring. As in [Ma], we first localize our ring at a multiplicative set, without changing the space of orders, in order to make the valuation theory work better. This is a standard idea from real algebraic geometry. Remarkably, many of the notions, methods, and results for fields carry over to this new setting. We define compatiblity between valuations and orders and preorders, and the ring A(T ) associated to a preorder T , which turns out to be Prufer ring as in the field case. We define the relation of dependency on the set of valuations associated to a preorder and we use this to prove a decomposition theorem for the space of orders. We can then apply this to show that, under a certain finiteness condition, the space of orders is equivalent to the space of orders of a preordered field.

Holomorphy rings and higher level orders on skew fields

The intimate connections which exist between higher level orders, sums of 2n th powers, valuation theory, and real-valued places on a field have been explored extensively. Craven initiated the study of similar notions for skew fields in [4]. In [12] we pursued these ideas, but the best results required a 2-primary assumption. This paper extends the work of [12]; in particular, we prove an “Artin-Schreier” theorem for preordered skew fields, namely that a preorder T in a skew field is the intersection of the kernels of all the signatures that contain T. This allows us to obtain the results of [12] without the 2-primary assumption. In particular, we have that preordered skew fields give rise to spaces of signatures, Mulcahy’s abstract setting for the theory of higher level forms and reduced Witt rings. Our main tools are valuation theory and the Kadison-Dubois representation theorem for partially ordered rings. As in the field case, we make use of the ring A, associated to a preorder T, which is the intersection of certain valuation rings. Along the way to our main theorem we prove some facts about noncommutative Priifer rings, and we show that A, is a Priifer ring. While our study of preordered skew fields follows along lines which are familiar from the field context, there are significant differences between the two theories. For example, there are skew fields which have no orders, but admit valuation rings with ordered residue fields.

Witt rings and almost free pro-2-groups

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

A structure theorem for abstract Witt rings containing rigid elements

Suppose ( R, G) is a non-reduced abstract Witt ring containing a rigid element d such that the value set D<1,- d> satisfies a certain finiteness-condition. Then ( R, G) is a direct product of a reduced abstract Witt ring and a Witt-group ring.

A representation theorem for higher level reduced Witt rings

In [6] Becker and Rosenberg defined the reduced Witt ring WAK) of a formally real field K, with respect to a higher level preorder Tin K. WAK) is a subring of the ring C(X,, @) of continuous c-valued functions on the space X, of signatures of K which are trivial on T, and is built up from higher level T-forms over K. This higher level reduced theory is a natural generalization of the reduced theory of quadratic forms over fields. Spaces of Signatures were introduced in [18] to provide an abstract framework for the higher level reduced theory over fields, along the lines of Marshall’s Spaces of Orderings [13-H] treatment of the classical reduced theory of quadratic forms. There are many advantages to this axiomatic approach. It allows for a unified study of reduced Witt rings over fields and skew fields (Powers has shown that Becker and Rosenberg’s study of higher level reduced Witt rings can also be carried out over skew fields [22]). The absence of technical considerations such as valuation theory not only yields simpler proofs of known results, but also new results about the theory over fields [ 183. The Space of Orderings version [ 151 of the result which we prove here has found application in real algebraic geometry, and we hope that our abstract higher level result might also prove useful in this context. The question which concerns us here is the so-called representation problem: given FE C(X, @), how can we tell if F=f for some f~ W(X)? This has been solved for the higher level reduced Witt ring of a field by Becker and Rosenberg [6] (see [S] also), and in the case of Spaces of Orderings by Marshall [15], both solutions being generalizations of the original representation theorems of Becker and Brlicker [4, 5.31 and Brown and Marshall [9, 4.21 (in the context of the reduced quadratic form theory over fields). Under a 2-power assumption we simultaneously extend all of these representation theorems to our abstract higher level setting. This entails studying fans and more importantly a generalization called quasifans. We not only obtain a common proof of the representation 105 0021-8693/88 $3.00

Characterizing reduced Witt rings of higher level

Characterizing reduced Witt rings of higher level

Reduced forms and reduced Witt rings of higher level

The present paper and its successor [S] deal with an extension of the socalled reduced theory of quadratic forms over a formally real field K [5, 7, lo]. This theory deals with the reduced Witt ring Wred(K) = W(K)/Nil(W(K)) and its natural embedding into C(X, Z), the ring of integer valued continuous functions on the space X = X(K) of orderings of K. Recently it has been found that there is a far-reaching generalization of the notion of ordering, namely, the orderings of higher level [ 11. The orderings of level s, where s divides a given natural number n, are related to sums of 2nth powers in the same way as the usual orderings are to sums of squares; their intersection is the set of all sums of 2nth powers in K [ 11. In [6] orderings of levels were replaced by signatures of this level. These are those characters x: I? + S’, of order 2s, whose kernels are additively closed, where k = K\{O}. It is exactly this change in point of view which leads to a natural definition of reduced Witt rings of higher level: Let X,, =X,(K) denote the set of all signatures of K whose order divides 2n, so that X,(K) =X(K) =X. Being a closed subspace of the compact character group of x, the set X,(K) is also a compact topological space. Each element a E I? induces a continuous function a^: X, -+ C in C(X,, C), defined by x++ x(a), where C is given the discrete topology. By analogy with the natural representation of the ordinary reduced Witt ring as a subring of 477 0021-8693/85 $3.00

The model theory of ‘ R-formal’ fields

Let K be a field, and let W( K) denote its Witt ring of Quadratic Forms. It is well-known in the theory of Quadratic Forms that the orders of K correpond in a one to one way with all ring surjections W(K) → Z . In particular, a field L is formally real over an ordered field K if and only if there is a homomorphism ϕ 1: W(L)→ Z which extends the given ‘signature’ ϕ K: W(K)→ Z . (E.g. ϕ K = ϕ 1, i ∗, where i ∗: W(K)1 → W(L) is the functinal map.) Using the above, one may discuss the usual theory of formally real and real closed fields in terms of Witt rings, Knebusch in [6] has, in the above setting, given a remarkable new proof of the uniqueness of real closures. One might ask what happens when the Z above is replaced by some other ring R? That is the subject of this present note. In particular, we shall prove some algebraic and model theoretic analogues of well-known results for real closed fields, where the above Z is replaced by some finitely generated reduced Witt ring.

An application of Pólya's theory of counting to an enumeration problem arising in quadratic form theory

In this paper we apply Pólya's theory of counting to compute the number of isomorphism types of reduced Witt rings of fields with a fixed finite number of orderings. The problem is first transformed into a graph theoretical enumeration problem involving unlabeled rooted trees with certain numbers assigned to the vertices.

The Witt ring of a space of orderings

The theory of “space of orderings” generalizes the reduced theory of quadratic forms over fields (or, more generally, over semilocal rings). The category of spaces of orderings is equivalent to a certain category of “abstract Witt rings". In the particular case of the space of orderings of a formally real field K, the corresponding abstract Witt ring is just the reduced Witt ring of K. In this paper it is proved that if X = ( X , G ) X\, = \,(X,\,G) is any space of orderings with Witt ring W(X), and X → Z X\, \to \,Z is any continuous function, then g is represented by an element of W(X) if and only if Σ σ ∈ V g ( σ ) ≡ 0 mod | V | {\Sigma _{\sigma \, \in \,V}}g(\sigma )\, \equiv \,0\,\bmod \,\left | V \right | holds for all finite fans V ⊆ X V \subseteq X . This generalizes a recent field theoretic result of Becker and Bröcker. Following the proof of this, applications are given to the computation of the stability index of X, and to the representation of continuous functions g : X → ± 1 g:\,X\, \to \, \pm 1 by elements of G.

Characterizing reduced Witt rings. II

Characterizing reduced Witt rings. II

Characterizing reduced Witt rings of fields

Characterizing reduced Witt rings of fields

The Reduced Witt Ring of a Formally Real Field

The Reduced Witt Ring of a Formally Real Field

The reduced Witt ring of a formally real field

The reduced Witt ring of a formally real field