Free pro-\(p\) product decompositions of Galois groups
Let \(p\) be a prime number, let \(K\) be a field of characteristic \(\neq p\) containing the roots of unity of order \(p\), and let \(G_K(p)\) be the maximal pro-\(p\) Galois group of \(K\). Suppose that \(K= \widehat K_1\cap \cdots\cap \widehat K_n\), where for \(1\leq i\leq n\), \(\widehat K_i\) is a subextension of \(K(p)/K\) which is either: (i) \(p\)-henselian with respect to a valuation with non-\(p\)-divisible value group; (ii) \(p\)-henselian with respect to a valuation with residue characteristic \(p\); or (in case \(p=2)\): (iii) a euclidean field. It is shown that when \(n\) is minimal, \(G_K(p)= G_{\widehat K_1} (p)*_p \cdots*_p G_{\widehat K_n} (p)\) (a free product in the category of pro-\(p\) groups). It is also shown that in most cases this decomposition is the finest possible. As applications one obtains local-global principles for Brauer groups and for quadratic forms over \(K\). The latter is a partial generalization of a local-global principle due to Bröcker and Prestel (independently) in the reduced theory of quadratic forms.
scientific article; zbMATH DE number 1033932