Definability of the ring of integers in pro-\(p\) Galois extensions of number fields
Let \(K\) be a number field and \(M\) a field of algebraic numbers (in a fixed algebraic closure of \(\mathbb Q\)) which extends \(K\). Let \(\widetilde {\mathbb Z}\) be the ring of all algebraic integers. The author proves that if \(M\) is pro-\(p\) for some prime number \(p\), then the ring of algebraic integers of \(M\), \(M\cap \widetilde {\mathbb Z}\), is first-order definable in \(M\), with parameters in \(M\). The methods of proof have their origin in previous relevant work of the author and builds on classical definability results of Rumely (who extended similar results of J. Robinson). The motivation of the paper is to investigate the decidability question for the fields \(M\). This relates to decidability of fields of algebraic numbers which relate to geometrical constructions (e.g. fields of complex or totally real numbers constructible by ruler and compass). The bibliography contains references on several such classical results. Due to results of Rumely, van den Dries and Macintyre on the decidability of the theory of \(\widetilde {\mathbb Z}\), the results of the paper do not suffice to show undecidability of the fields \(M\). It seems strange that, even today, many of the fundamental and intriguing questions which motivate this line of research, are still open.
numbers constructible by rulers and compass
scientific article; zbMATH DE number 1506579