Hereditarily quadratically closed fields
type: Item
instance of: scholarly article
title: Hereditarily quadratically closed fields
zbMATH Open document ID: 0669.12010
DOI: 10.1016/0021-8693(87)90231-6
author: Richard Elman
author: A. R. Wadsworth
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The authors prove for a finitely generated field extension K/F with \(tr.\deg_ FK=n\geq 1\) and char(F)\(\neq 2\) the equivalence of the following statements: (1) F(\(\sqrt{-1})\) and all its algebraic extensions are quadratically closed; (2) \(cd_ 2K(\sqrt{-1})=n\), where \(cd_ 2(F)\) means the cohomological 2-dimension of the absolute Galois group of the field F; (3) \(u(K(-1))=2^ n\) for \(u(F)=:\sup \{\dim (q)| \quad q\) is an anisotropic F-quadratic form with zero signature in each real closure of \(F\}\) ; (4) \(I^{n+1}K\) ( \(= the\) ideal in the Witt ring of K generated by the n-fold Pfister forms, i.e. forms of type \(\otimes^{n}_{i=1}<1,a_ i> )\) contains no \(n+1\)-fold anisotropic Pfister form having zero signature in each real closure of K; (5) \(I^{n+1}K\) is torsion-free; (6) min\(\{\) \(r|\) \(I^{r+1}K=2I^ rK\} \leq n\).
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 4095624
zbMATH Keywords: quadratically closed field
zbMATH Keywords: quadratic form
zbMATH Keywords: Pfister form
reviewed by: Alexander Prestel
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90231-6
OpenAlex ID: W2074863113
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description: scientific article; zbMATH DE number 4095624