Definability and decidability for rings of integers in totally imaginary fields

Abstract

AbstractWe show that the ring of integers of is existentially definable in the ring of integers of , where denotes the field of all totally real numbers. This implies that the ring of integers of is undecidable and first‐order nondefinable in . More generally, when is a totally imaginary quadratic extension of a totally real field , we use the unit groups of orders to produce existentially definable totally real subsets . Under certain conditions on , including the so‐called ‐number of being the minimal value , we deduce the undecidability of . This extends previous work that proved an analogous result in the opposite case . In particular, unlike prior work, we do not require that contains only finitely many roots of unity.