On the cohomological characterization of real free pro-\(2\)-groups
A real free pro-2-group is a free pro-2-product of copies of \(\mathbb{Z}/2 \mathbb{Z}\) with a free pro-2-group. The main result of this paper is the following cohomological characterization: a pro-2-group \(G\) is real free if and only if \(G\) admits a free closed subgroup \(H_0\) of index 2 such that the corresponding quadratic character \(\chi_0\) satisfies \(\chi_0 \cup \chi = \chi \cup \chi\) for every \(\chi \in H^1 (G, \mathbb{Z}/2 \mathbb{Z})\). In particular, for any field \(K\) of characteristic \(\neq 2\), the Galois group \(G_K(2)\) of the maximal pro-2-extension of \(K\) is real free if and only if its closed subgroup \(H_0\) corresponding to \(K (\sqrt {-1})\) is free (the additional condition on the character \(\chi_0\) is automatically verified because of classical properties of quaternion algebras). This gives an ample provision of examples of Galois groups which are real free pro-2-groups. Recall that the Hasse invariant \(\widetilde u(K)\) is the maximal dimension of the anisotropic quadratic forms over \(K\) which become isotropic over all (if any) real closures of \(K\). As an application of his criterion, the author proves that for a formally real field \(K\), the following conditions are equivalent: i) \(G_K (2)\) is real free; ii) \(\widetilde u(K) \leq 2\); iii) \(I^2 K = 2IK\) and every sum of squares is a sum of two squares. Here \(IK\) is the usual fundamental ideal of the Witt ring of \(K\).
scientific article; zbMATH DE number 866166