Rigid elements, valuations, and realization of Witt rings
type: Item
instance of: scholarly article
title: Rigid elements, valuations, and realization of Witt rings
zbMATH Open document ID: 0629.10016
DOI: 10.1016/0021-8693(87)90057-3
author: Jón Kr. Arason
author: Richard Elman
author: Bill Jacob
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let T be a subgroup of the multiplicative group \(F^{\times}\) of a field F. Then \(a\in F^{\times}\) is said to be T-rigid, if \(T+aT\subseteq T\cup aT\). It turns out that T-rigid elements in F give rise to a valuation of the field F. This was first discovered by the third author [ibid. 68, 247-267 (1981; Zbl 0457.10007)] in case F is formally real and \(F^{\times 2}\subseteq T\) and then generalized to arbitrary fields by \textit{R. Ware} [Can. J. Math. 33, 1338-1355 (1981; Zbl 0514.10015)] still assuming \(F^{\times 2}\subseteq T.\) In the present paper these results are extended to cover the case where the squares do not necessarily sit in T. A section is devoted to adjusting the results to applications in quadratic form theory. Then the authors prove the following important realization theorem for Witt rings. If the Witt ring WF of a field F of characteristic not two is isomorphic to the product \(R\times S\) of two abstract Witt rings and if R is not basic (i.e. the abstract square class group \(G_ R\) contains birigid elements), then there is a 2-extension K of F such that the Witt ring WK is isomorphic to R. Moreover, the two isomorphisms, the natural morphism WF\(\to WK\) and the projection of \(R\times S\) onto R, combine into a commutative diagram. Application of the realization theorem in a forthcoming paper by the authors is announced. \{Another application to determining all profinite 2-groups which occur as the Galois groups of the maximal 2-extensions of fields with finite number of square classes can be found in a preprint by the third author and \textit{R. Ware}, ``Elementary Witt rings and pro-2 Galois groups'' (1987).\}
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 12F99
zbMATH DE Number: 4023378
zbMATH Keywords: valuation
zbMATH Keywords: rigid elements
zbMATH Keywords: realization theorem for Witt rings
zbMATH Keywords: characteristic not two
zbMATH Keywords: isomorphisms
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90057-3
OpenAlex ID: W1967710359
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description: scientific article; zbMATH DE number 4023378