A survey of undecidability problems of rings of totally real algebraic integers (Model theoretic aspects of the notion of independence and dimension)

Abstract

Let mathbb{Z}^{tr} be the ring of all totally real algebraic integers in mathbb{C}. We consider (un)decidability of its subrings of infinite degree over mathbb{Q}. Julia Robinson [Ro] proved that mathbb{Z} is first order definable (without parameters) in mathbb{Z}^{tr}, thus showed that it is undecidable. Moreover she showed undecidability of the rings of (algebraic) integers of any subfield of mathbb{Q} ({sqrt{p}|p prime}) also by showing the definability of mathbb{Z} in those rings. From her remark in [Ro], it seems that we may conjecture that all subrings of mathbb{Z}^{tr} are undecidable. We survey recent progress on this problem. We note that rings of algebraic integers of finite degree over mathbb{Q} are undecidable. This fact is also proved in [Ro].