  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.
