Defining $\mathbb Z$ using unit groups

Abstract

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\Q$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order definition of $\Z$. Namely, we prove that for a large collection of algebraic extensions $K/\Q$, $$ \{x \in \oo_K : \text{$\forall \e \in \oo_K^\times \;\exists \delta \in \oo_K^\times$ such that $\delta-1 \equiv (\e-1)x \pmod{(\e-1)^2}$}\} = \Z $$ where $\oo_K$ denotes the ring of integers of $K$. One of the corollaries of our results is undecidability of the field of constructible numbers, a question posed by Tarski in 1948.