Alex Rosenberg

Structure of Witt rings, quotients of abelian group rings and orderings of fields

In particular, Theorems 5 and 6 yield the results of [5, §3] for Witt rings of formally real fields and Theorem 7 those of [5, §5] for Witt rings of nonreal fields. By studying subrings of the rings described in Theorems 5-7 and using the results of [2] for symmetric bilinear forms over a Dedekind ring

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

Sums of Squares in Some Integral Domains

3.00

A theorem of Harrison, Kummer theory, and Galois algebras

Let A be an integral domain, F q f ( A ) its quotient field; let a be an element of A . I f a is a sum of n squares in F, is a then a sum of squares in A , and, i f so, is a a sum of n squares in A ? As may be expected, the answers to these questions depend on the number n and on the nature of the domain A. The following are some classical examples: (1) If n = 1 and A is integrally closed, the answers to both questions are yes. (2) If A = 71, the answers to both questions are yes, for any n. This follows from classical number theory (e.g., [27, Chap. 7]).

Signatures on semilocal rings

Let R be a field and S a separable algebraic closure of R with galois group R . In [8] Harrison succeeded in describing R / ′ R in terms of R only. More precisely, he constructed a certain complex ( R, Q/Z ) and proved Hom c , where Hom c denotes continuous homomorphisms and H 2 stands for the second cohomology group of the complex . In this paper, which is mainly expository in nature, we reexamine Harrison’s proof and show how [8] connects with Kummer theory and the theory of galois algebras [16]. We emphasize that most of the ideas on which this paper is based originate in [8].

The Global Dimensions of Ore Extensions and Weyl Algebras

Most of the results of this paper have been announced in [31, Section 3] and, in slightly simplified form, in [32]. The reader is advised to consult these announcements for an outline of the contents of the present work. One of our main purposes here is to extend part of the Artin-Schreier theory of real closed fields to commutative semilocal rings with involution. The central concept that enables us to accomplish this goal is that of a signature: Let C denote a semilocal ring with an involution / whose fixed ring we denote by A. A signature is a homomorphism a from A*, the group of units of A, to {±1} with certain properties (Definition 2.1 and Proposition 2.4). If A is a field the signatures correspond byectively with the set of total orderings of A for which all the norms N(c) = c J(c) for c in C * are positive. In particular then, if / is the identity, this latter set consists of all orderings of A. In Section 2 we study the notion of signature. By definition a signature a on (C, /) corresponds with a unique homomorphism 5 from the Witt ring WF(C, J) of free Hermitian spaces over (C, /) to Z and conversely. For the case / the identity and C = A a field this correspondence between orderings

Global Dimension of Ore Extensions

This chapter discusses the global dimensions of ore extensions and Weyl algebras. If R is a ring and D is a derivation of R , and if S = R[t, D] is the ore extension of R with respect to D , that is, S is additively the group of polynomials in an indeterminate t with multiplication subject to tr = rt + D(r) for all r in R , then an extension of D to a derivation of S by setting D(t) = 0 for all elements s of S leads to the result ts = st + D(s). If R is a commutative noetherian ring, and if M → 0 is a left A 1 R -module that is finitely generated as a left R -module, then M is an abelian torsion group.

Galois theory and Galois cohomology of commutative rings

This chapter discusses the global dimension of ore extensions. If R is a left and right noetherian and of a finite left global dimension, a necessary condition for equality to hold is the existence of a left S -module M that is finitely generated as an R -module and with l.dim R M = l.gl.dim R . If R is a ring with derivation D , then S = R [ t ] is the ore extension of R with respect to D , that is, S is additively the group of polynomials in an indeterminate t with multiplication subject to tr = rt + D(r) for all r in R . The chapter presents a situation in which R is commutative.