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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.02521 |
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| _version_ | 1866916271933620224 |
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| author | Mazur, Barry Rubin, Karl Shlapentokh, Alexandra |
| author_facet | Mazur, Barry Rubin, Karl Shlapentokh, Alexandra |
| contents | 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 $K/\Q$, $$ \{x \in \oo_K : \text{$\forall \e \in \oo_K^\times \;\exists δ\in \oo_K^\times$ such that $δ-1 \equiv (\e-1)x \pmod{(\e-1)^2}$}\} = \Z $$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_02521 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Defining $\mathbb Z$ using unit groups Mazur, Barry Rubin, Karl Shlapentokh, Alexandra Number Theory Logic 11U05 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 $K/\Q$, $$ \{x \in \oo_K : \text{$\forall \e \in \oo_K^\times \;\exists δ\in \oo_K^\times$ such that $δ-1 \equiv (\e-1)x \pmod{(\e-1)^2}$}\} = \Z $$ 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. |
| title | Defining $\mathbb Z$ using unit groups |
| topic | Number Theory Logic 11U05 |
| url | https://arxiv.org/abs/2303.02521 |