Definability and decidability for rings of integers in totally imaginary fields

TL;DR

This paper proves that rings of integers in certain imaginary quadratic extensions are undecidable and definable, extending previous results by removing restrictions on roots of unity and using unit groups to establish definability.

Contribution

It introduces new methods to show undecidability of rings of integers in totally imaginary fields without restrictions on roots of unity, expanding the scope of prior work.

Findings

Ring of integers of $ ext{Q}^{tr}$ is existentially definable in its quadratic extension.

Undecidability of rings of integers in certain totally imaginary quadratic fields.

New techniques using unit groups to define totally real subsets.

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.