The elementary theory of \(e\)-free PAC domains
In the author's words ``We prove that the theory of all sentences in the language of rings which are true in \({\widetilde {\mathbf Z}}\cap {\widetilde {\mathbf Q}}(\sigma)\) for almost all \(\sigma\) is decidable.'' Here \({\widetilde {\mathbf Q}}\) is the field of algebraic numbers, \({\widetilde {\mathbf Z}}\) is the ring of algebraic integers, \(\sigma\) is an element of the absolute Galois group of \({\mathbf Q}\) and \({\widetilde {\mathbf Q}}(\sigma)\) is the subfield of \({\widetilde {\mathbf Q}}\) which is fixed by \( \sigma\); the phrase ``almost all'' is meant in the sense of the Haar measure. This result should be compared with the results of Rumely, Van den Dries and Macintyre, according to which, the theory of \({\widetilde {\mathbf Z}}\) is decidable. In the opposite direction, due to work of J. Robinson, the theory of the ring of totally real algebraic integers is undecidable. The methods have their origin in work on Pseudo Algebraically Closed (PAC) fields, presented in the book ``Field arithmetic'' [Berlin etc.: Springer (1986; Zbl 0625.12001)] by \textit{M. D. Fried} and \textit{M. Jarden}.
scientific article; zbMATH DE number 1448518
scientific article