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. \end{abstract}
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Error Detection Architectures for Ring Polynomial Multiplication and Modular Reduction of Ring-LWE in ${\frac {\mathbb {Z}/p \mathbb {Z}[x]}{x^{n}+1}}$ Benchmarked on ASIC
- Award ID(s):
- 1801488
- PAR ID:
- 10180240
- Date Published:
- Journal Name:
- IEEE Transactions on Reliability
- ISSN:
- 0018-9529
- Page Range / eLocation ID:
- 1 to 9
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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