Finitely generated pro-\(p\) Galois groups of \(p\)-Henselian fields
Let \(p\) be a prime number and let \(K\) be a field of characteristic 0 containing the group \(\mu_p\) of all \(p\)-th roots of unity. Let \(v\) be a \(p\)-henselian Krull valuation of \(K\) with residue class field of characteristic \(p\). Assume that the maximal pro-\(p\) Galois group \(G_K(p)\) of \(K\) is finitely generated. The author proves that \(G_K(p)\) is a semi-direct product \({\mathbb Z}^m_p \propto G\), with suitable non-negative integer \(m\) where either (i) \(G\simeq G_{K^*}(p)\) for some finite extension \(K^*\) of \({\mathbb Q}_p(\mu_p)\), or (ii) \(G\) is a finitely generated free pro-\(p\) group. A similar characterization of the structure of finite generated pro-\(p\) absolute Galois groups of henselian fields with residue characteristic \(p\) is given. These results generalize results on finite extensions of \({\mathbb Q}_p\) obtained by Demushkin, Serre and Labute. This result is applied to show that if \(p=2\) and \(K\) satisfies the above assumptions, then the Witt ring \(W(K)\) is of elementary type.
scientific article; zbMATH DE number 1308370
scientific article