Demuškin fields with valuations
Let \(p\) be a fixed prime and \(q = p^s\). A pro-\(p\) group \(G\) is a \(q\)-Demushkin group if the profinite cohomology group \(H^1(G,{\mathbb Z}/q)\) is a finite free \({\mathbb Z}/q\)-module, \(H^2(G,{\mathbb Z}/q) \cong {\mathbb Z}/q\), and the cup product \(H^1(G,{\mathbb Z}/q) \times H^1(G,{\mathbb Z}/q) \to H^2(G,{\mathbb Z}/q)\) is nondegenerate. Now let \(F\) be a field of characteristic not \(p\), and let \(G_F(p)\) denote the maximal pro-\(p\) Galois group of \(F\). \(F\) is said to be a \(q\)-Demushkin field if \(G_F(p)\) is a \(q\)-Demushkin group. A field \(F\) is arithmetically \(q\)-Demushkin if there exists a valuation \(u\) on \(F\) with \(p\)-divisible value group and a discrete rank-1 valuation \(w\) on the residue field \(E = \overline F_u\) such that \(\operatorname{char} E = 0, \operatorname{char} \overline E_w = p, 1 - \mathfrak{m}_u \leq (F^\times)^q\), and \(1 - q^2\mathfrak{m}_w \leq (E^\times)^q\). In this case, if \(F\) contains the \(q\)th roots of unity, then \(F\) is in fact \(q\)-Demushkin of \(q\)-rank \(\geq 3\). This article provides affirmative answers to the following two open questions, under the additional assumption that \(F\) is equipped with a valuation \(v\) whose value group is not \(p\)-divisible, and whose decomposition field in \(G_F(p)\) does not contain all roots of unity of \(p\)-power order. (1) Are all \(q\)-Demushkin fields of \(q\)-rank \(\geq 3\) and containing the \(q\)th roots of unity arithmetically \(q\)-Demushkin? (2) Is every pro-\(p\) Demuškin group which is isomorphic to \(G_F(p)\) for some field \(F\) in fact isomorphic to \(G_F(p)\) for some \(p\)-adic field \(F\)?
scientific article; zbMATH DE number 1922384
scientific article