Defining $\mathbb Z$ using unit groups

Barry Mazur, Karl Rubin, Alexandra Shlapentokh

Published 2023-03-04, updated 2023-05-07Version 2

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\mathbb Q$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order definition of $\mathbb Z$. Namely, we prove that for a large collection of algebraic extensions $K/\mathbb Q$, $$ \{x \in {\mathcal O}_K : \text{$\forall \varepsilon \in {\mathcal O}_K^\times \;\exists \delta \in {\mathcal O}_K^\times$ such that $\delta-1 \equiv (\varepsilon-1)x \pmod{(\varepsilon-1)^2}$}\} = \mathbb Z $$ where ${\mathcal O}_K$ denotes the ring of integers of $K$.