Toward a theorem of Douady (Q541984)

Let \(F\) be an algebraically closed field of zero characteristic, and let \(\Phi = F(X)\) be a rational function field in one variable over \(F\). As shown by \textit{A. Douady} (see [C. R. Acad. Sci., Paris 258, 5305--5308 (1964; Zbl 0146.42105)]), then the absolute Galois group \(G _ {\Phi }\) is a free profinite group. For that purpose, he has first proved this theorem in the special case where \(F = \mathbb C\), the field of complex numbers (relying on Riemann's existence theorem), and thereafter he has extended it to the general case, making use of Grothendieck's transition method. Alternative transitional techniques relying on model theoretic methods have been used by van den Dries and Ribenboim, for proving Douady's theorem as well as for reproving the Krull-Neukirch theorem concerning \(\mathcal{G}_ {\mathbb R(X)}\) (see \textit{W. Krull} and \textit{J. Neukirch} [Math. Ann. 193, 197--209 (1971; Zbl 0236.12104) and, e.g., \textit{L. van den Dries} and \textit{P. Ribenboim}, Ann. Pure Appl. Logic 33, 83--107 (1987; Zbl 0645.12009)]). The paper under review presents a new model-theoretic transition method, which may also be helpful in describing \(\mathcal{G}_{R(X)}\), where \(R\) is a real closed field. The models considered by the author satisfy a system of elementary conditions of four different types.