Fields Galois-equivalent to a local field of positive characteristic
type: Item
instance of: scholarly article
title: Fields Galois-equivalent to a local field of positive characteristic
zbMATH Open document ID: 0965.11048
author: Ido Efrat
author: I. B. Fesenko
published in: Mathematical Research Letters
publication date: 2000-08-13T00:00:00Z
review text: Let \(\mathcal F = F_{p^n}((t))\) be a non-archimedean local field of positive characteristic \(p\), and let \(K\) be any fields with absolute Galois group \(G_K\) isomorphic to the absolute Galois group \(G_{\mathcal F}\) of \(\mathcal F\). Then it is shown that \(K\) possesses a Henselian valuation \(v\) such that (1) the value group \(\Gamma\) of \(v\) satisfies \(\Gamma/l \cong Z/l\) for all prime numbers \(l \neq p\); (2) the residue field \(\overline K\) of \(v\) has characteristic \(p\); (3) the maximal prime-to-\(p\) Galois group \(G_{\overline K}(p')\) of \(\overline K\) is \(\widehat Z/ Z_p\); (4) if \(\text{char} K = 0\), then \(\Gamma = p\Gamma\) and \(\overline K\) is perfect. For every positive integer \(r\), the authors construct such fields \(K\) of characteristic \(p\) with \(\Gamma/p \cong (Z/p)^r\). Likewise, examples are constructed with \(\Gamma \cong Z, G_{\overline K} \not\cong \widehat Z\), and \(\overline K\) imperfect. These results give a non-archimedean analogue to the celebrated result of Artin and Schreier, characterizing fields whose absolute Galois group is isomorphic to that of the real numbers as the real closed fields. The similar problem for \(p\)-adic fields has been answered independently by the first author [Isr. J. Math. 91, 273-284 (1995; Zbl 0842.12003)] and by \textit{J. Koenigsmann} [J. Reine Angew. Math. 465, 165-182 (1995; Zbl 0824.12006)]: the fields whose absolute Galois groups are isomorphic to the absolute Galois group of some finite extension of \(\mathbb{Q}_p\) are precisely the \(p\)-adically closed fields in the sense of \textit{A. Prestel} and \textit{P. Roquette} [Formally \(p\)-adic fields, Lect. Notes Math. 1050 (Springer, Berlin, 1984; Zbl 0523.12016)].
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 19F99
Mathematics Subject Classification ID: 12F12
zbMATH DE Number: 1489499
zbMATH Keywords: Galois groups
zbMATH Keywords: local fields
zbMATH Keywords: valuation theory
MaRDI profile type: Publication
full work available at URL: mrl.1999.v6.n3.a8
OpenAlex ID: W1988707188
DOI: 10.4310/MRL.1999.V6.N3.A8
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description: scientific article; zbMATH DE number 1489499