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Formally real fields, pythagorean fields, C-fields and W-groups. (English) Zbl 0692.12005

In previous work the authors showed that under some mild condition on the field F the Galois group \({\mathcal G}_ F\) of a certain 2-extension \(F^{(3)}\) of F completely determines the Witt ring of F. In this paper it is shown that the presence or absence of certain involutions in \({\mathcal G}_ F\) determines whether F is a formally real field or not. More precisely a 1-1 correspondence is shown between orderings of fields and certain classes of involutions in \({\mathcal G}_ F\). The relative real closure of a formally real field F inside of \(F^{(3)}\) is defined and this is used to give a new characterization of superpythagorean fields. Moreover a characterization of pythagorean fields via \({\mathcal G}_ F\) is shown as well as the description of all possible Abelian groups \({\mathcal G}_ F\). The main tools are dihedral and \({\mathbb{Z}}/4 {\mathbb{Z}}\) extensions, together with simple Galois theory.
This paper can be viewed as transfer of classical results of Artin- Schreier and Becker to much smaller fields.
Reviewer: J.Mináč

MSC:

12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
12J15 Ordered fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
12F10 Separable extensions, Galois theory
11E10 Forms over real fields
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References:

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