A Galois-theoretic characterization of \(p\)-adically closed fields
Let \(p\) be an odd prime and \(K\) a field. It follows from results of [\textit{A. Prestel} and \textit{P. Roquette}, Formally \(p\)-adic fields, Lect. Notes Math. 1050 (Springer, Berlin, 1984; Zbl 0523.12016)]\ that if \(K\) is \(p\)-adically closed, then its absolute Galois group \(G_K\) is an open subgroup of \(G_{\mathbb{Q}_p}\). It was conjectured that the converse also holds. In [\textit{F. Pop}, J. Reine Angew. Math. 392, 145-175 (1988; Zbl 0671.12005)]\ it was shown that the conjecture is true if \(K\) is henselian with respect to a valuation having positive residue characteristic. The main result of the present paper is a positive answer to the above conjecture for \(p\neq 2\). The idea is to use a valuation-theoretic construction of [\textit{J. K. Arason}, \textit{R. Elman} and \textit{B. Jacob}, J. Algebra 110, 449-467 (1987; Zbl 0629.10016)]\ to show that one can endow \(K\) with a henselian valuation with positive residue characteristic, then apply the result of [\textit{F. Pop}, loc. cit]. By the same method, the author shows that if \(q\) is an odd prime power and \(K\) is a field such that \(G_K \simeq G_{\mathbb{F}_q ((t))}\), then \(K\) is henselian with respect to a non-trivial valuation. In [\textit{J. Koenigsmann}, J. Reine Angew. Math. 465, 165-182 (1995; Zbl 0824.12006)], an independent proof of the main result is given, which covers also the case \(p=2\).
scientific article; zbMATH DE number 820103
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