Defining $\mathbb{Z}$ using unit groups

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\text{Q}$, 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 $K\mathrm{/}\text{Q}$, $$\{x\in {\text{oo}}_K:\mathrm{\forall }\text{e}\in {\text{oo}}_K^{\times }\mathrm{\exists }\delta \in {\text{oo}}_K^{\times }\text{ such that }\delta -1\equiv (\text{e}-1)x(\mathrm{m}\mathrm{o}\mathrm{d}(\text{e}-1{)}^2)\}=\mathbb{Z}$$ where ${\text{oo}}_K$ 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.