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Transactions of the American Mathematical Society

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On Diophantine definability and decidability in some infinite totally real extensions of $\mathbb Q$


Author: Alexandra Shlapentokh
Journal: Trans. Amer. Math. Soc. 356 (2004), 3189-3207
MSC (2000): Primary 11U05, 11U09; Secondary 03C07
DOI: https://doi.org/10.1090/S0002-9947-03-03343-9
Published electronically: November 4, 2003
MathSciNet review: 2052946
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Abstract: Let $M$ be a number field, and $W_M$ a set of its non-Archimedean primes. Then let $O_{M,W_M} = \{x \in M\vert \operatorname{ord}_{\mathfrak{t}}x \geq 0, \, \forall \mathfrak{t} \, \not \in W_M\}$. Let $\{p_1,\ldots,p_r\}$ be a finite set of prime numbers. Let $F_{inf}$ be the field generated by all the $p_i^{j}$-th roots of unity as $j \rightarrow \infty$ and $i=1,\ldots,r$. Let $K_{inf}$ be the largest totally real subfield of $F_{inf}$. Then for any $\varepsilon > 0$, there exist a number field $M \subset K_{inf}$, and a set $W_M$ of non-Archimedean primes of $M$ such that $W_M$ has density greater than $1-\varepsilon$, and $\mathbb{Z} $ has a Diophantine definition over the integral closure of $O_{M,W_M}$ in $K_{inf}$.


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Additional Information

Alexandra Shlapentokh
Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
Email: shlapentokha@mail.ecu.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03343-9
Keywords: Hilbert's tenth problem, Diophantine definability
Received by editor(s): June 5, 2000
Received by editor(s) in revised form: March 10, 2003
Published electronically: November 4, 2003
Additional Notes: The research for this paper has been partially supported by NSA grant MDA904-98-1-0510 and NSF grant DMS-9988620
Article copyright: © Copyright 2003 American Mathematical Society