On virtually projective groups
Let \(G_K (2)\) denote the maximal pro-2 Galois group of a field \(K\). Fields \(K\) for which \(G_{K\sqrt {-1}} (2)\) is a free pro-2 group are studied in the paper. In sections 2-4 the author extends the structure theory of such fields and generalizes some of their Galois-theoretic properties to a purely profinite group-theoretic setting. Next the paper is concerned with the following structure theorem proved by \textit{Yu. Ershov} [Math. Notes 36, 956-961 (1984); translation from Mat. Zametki 36, 913-923 (1984; Zbl 0573.12008)] using a field theoretic characterization of such groups by quadratic forms due to \textit{R. Ware} [J. Algebra 58, 227-237 (1979; Zbl 0412.10010)]: \(G_{K\sqrt {-1}} (2)\) is a free pro-2 group if and only if \(G_K (2)\) is a free pro-2 product of groups of order 2 and of a free pro-2 group. This theorem was generalized by \textit{D. Haran} [J. Algebra 156, 219-236 (1993; Zbl 0829.20042)] to an arbitrary pro-2 group using a cohomology theory of Artin-Schreier structures as follows: If \(G\) has a free pro-2 subgroup of index 2 and every involution of \(G\) is self-centralized, then \(G\) is a free pro-2 product of groups of order 2 and of a free pro-2 group. In section 5 the author gives a simplified proof of Ershov's theorem using only standard methods of Galois cohomology. Note that the problem of describing virtually free pro-\(p\) groups is open even in the finitely generated case, and there are only a few published results concerning it.
scientific article; zbMATH DE number 881133