Quaternion fields inside Pythagorean closure
type: Item
instance of: scholarly article
title: Quaternion fields inside Pythagorean closure
zbMATH Open document ID: 0678.12009
DOI: 10.1016/0022-4049(89)90029-7
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: In J. Pure Appl. Algebra 30, 95-107 (1983; Zbl 0518.10024)], \textit{R. Ware} proved the following theorem: If F is a formally real field and \(G=G_ F(py)\) is the Pythagorian Galois group of F (i.e. the Galois group of the smallest algebraic extension, F(py), of F in which every sum of squares is a square). Then the following properties are equivalent: (1) Every extension of F, of degree 4, contained in F(py), is normal over F. (2) The dihedral group of order 8 is not a homomorphic image of G. In the paper the author shows that actually (1) and (2) are also equivalent to the statement: Every homomorphic image of degree \( 8\) of the group G is abelian.
reviewed by: Jan Van Geel
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 12F05
zbMATH DE Number: 4110816
zbMATH Keywords: formally real field
zbMATH Keywords: Pythagorian Galois group
zbMATH Keywords: sum of squares
Wikidata QID: Q114215394
author: Ján Mináč
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90029-7
OpenAlex ID: W2019637103
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description: scientific article; zbMATH DE number 4110816