Fields Galois-equivalent to a local field of positive characteristic
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)].
scientific article; zbMATH DE number 1489499