Definability and undecidability via the torsion subgroup of units

Abstract

In this paper, we prove that $\mathbb{Z}$ is first-order definable in the ring of integers $\mathbb{Z}^{\text{ab}}$ of the maximal abelian extension $\mathbb{Q}^{\text{ab}}$ of $\mathbb{Q}$, which implies that the first-order theory of $\mathbb{Z}^{\text{ab}}$ is undecidable. More generally, writing $i = \sqrt{-1}$ and $\mathbb{Q}^{\text{tr}}$ for the field of all totally real numbers, we prove new definability and undecidability results for rings of integers of subfields of $\mathbb{Q}^{\text{tr}}(i)$, focusing especially on fields which contain infinitely many roots of unity. The key ingredient for these results is that there is a parameter-free positive-existential formula which defines the roots of unity $μ(\mathcal{O}_L)$ inside $\mathcal{O}_L$ for every field $L\subseteq \mathbb{Q}^{\text{tr}}(i)$.