An application of Tarski's principle to absolute Galois groups of function fields
In this paper the authors reprove Douady's result that the absolute Galois group of the rational function field C(t) over an algebraically closed field C of characteristic zero is the free profinite group generated by the set C. In the paper of \textit{A. Douady} [C. R. Acad. Sci., Paris 258, 5305-5308 (1964; Zbl 0146.421)] the essential ingredient is the `change of base field' from \({\mathbb{C}}\) to C provided by Grothendieck's `profinite fundamental group functor'. The authors instead use model theoretic methods in order to transfer the classical result for \({\mathbb{C}}(t)\) to C(t). They also treat in a similar manner the case of the absolute Galois group of the rational function field R(t) where R is any real closed field. They use model theoretic methods (like the Tarski principle) in order to transfer the result for \({\mathbb{R}}(t)\) [obtained by \textit{W. Krull} and \textit{J. Neukirch} in Math. Ann. 193, 197-209 (1971; Zbl 0236.12104)] to the case of R(t).
absolute Galois group of the rational function field
scientific article; zbMATH DE number 4051762
scientific article