Definability and decidability for rings of integers in totally imaginary fields

Abstract

We show that the ring of integers of $\mathbb{Q}^{\text{tr}}$ is existentially definable in the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$, where $\mathbb{Q}^{\text{tr}}$ denotes the field of all totally real numbers. This implies that the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$ is undecidable and first-order non-definable in $\mathbb{Q}^{\text{tr}}(i)$. More generally, when $L$ is a totally imaginary quadratic extension of a totally real field $K$, we use the unit groups $R^\times$ of orders $R\subseteq \mathcal{O}_L$ to produce existentially definable totally real subsets $X\subseteq \mathcal{O}_L$. Under certain conditions on $K$, including the so-called JR-number of $\mathcal{O}_K$ being the minimal value $\text{JR}(\mathcal{O}_K) = 4$, we deduce the undecidability of $\mathcal{O}_L$. This extends previous work which proved an analogous result in the opposite case $\text{JR}(\mathcal{O}_K) = \infty$. In particular, unlike prior work, we do not require that $L$ contains only finitely many roots of unity.