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The absolute Galois group of a pseudo real closed field. (English) Zbl 0595.12013
The authors give a characterization of the absolute Galois group of prc- fields. A field K is called prc if each K-variety which has a simple point in every real closure of K admits a K-rational point. The authors call a profinite group G ’real projective’ if for all finite groups A,B and all epimorphisms \(\alpha\) : \(B\to A\) and \(\phi\) : \(G\to A\) such that to every involution \(g\in G\) with \(\phi\) (g)\(\neq 1\) there exists an involution \(b\in B\) satisfying \(\alpha (b)=\phi (g)\), we can find a homomorphism \(\gamma\) : \(G\to B\) such that \(\alpha \circ \gamma =\phi.\)
The main result of the paper is that the absolute Galois group of a prc- field is real projective, and that every real projective profinite group is the absolute Galois group of some prc-field. In order to prove this deep result, the authors introduce the category of Artin-Schreier structures and define projectivity in such a way that the ’underlying’ profinite groups of projective Artin-Schreier structures are exactly the real projective ones.
Reviewer: A.Prestel

MSC:
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
14G05 Rational points
12F10 Separable extensions, Galois theory
20E18 Limits, profinite groups
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