# Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator

## Authors

DOI: 10.1007/s002220050170

- Cite this article as:
- Shlapentokh, A. Invent math (1997) 129: 489. doi:10.1007/s002220050170

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## Abstract.

Let *K* be a number field. Let *W* be a set of non-archimedean primes of *K*, let *O*_{K}, _{W}={*x*∈*K*∣*ord*_{p}*x*≥0∀*p*∉*W*}. Then if *K* is a totally real non-trivial cyclic extension of ℚ, there exists an **infinite** set *W* of finite primes of *K* such that ℤ and the ring of algebraic integers of *K* have a Diophantine definition over *O*_{K}, _{W}. (Thus, the Diophantine problem of *O*_{K}, _{W} is undecidable.)

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© Springer-Verlag Berlin Heidelberg 1997