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Galois groups of maximal 2-extensions. (English. Russian original) Zbl 0573.12008
Math. Notes 36, 956-961 (1984); translation from Mat. Zametki 36, No. 6, 913-923 (1984).
Let \(G_ F(2)\) and \(G_{\pi}\) denote the Galois group of the quadratic and Pythagorean closure of the field F, respectively. We say that a formally real field F satisfies \(H_ 4\) if every totally indefinite form over F of dimension 4 is isotropic. It is proved that \(G_ 2(F)\) of a Pythagorean SAP-field is isomorphic to the (topological) free \(X_ F\)- power of the two element cyclic group \({\mathbb{Z}}/2\) (i.e. \(G_ F(2)\cong ({\mathbb{Z}}/2)^{[X_ F]}\), where \(X_ F\) is the set of orderings of F.
The Galois group \(G_ F(2)\) of an arbitrary field satisfying \(H_ 4\) is isomorphic to the free wreath product of \(({\mathbb{Z}}/2)^{[X_ F]}\) and \(G_{\pi}\). Using previously obtained results the author shows that the Brauer group of a pseudo-real closed field F is isomorphic to the group of continuous functions \(C(X_ F, {\mathbb{Z}}/2)\).
Reviewer: M.Kula

12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
11E10 Forms over real fields
12G05 Galois cohomology
14F22 Brauer groups of schemes
12F10 Separable extensions, Galois theory
Full Text: DOI
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