Inventiones mathematicae

, Volume 129, Issue 3, pp 489–507

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


  • Alexandra Shlapentokh
    • Department of Mathematics, East Carolina University, Greenville, NC 27858, USA (e-mail:

DOI: 10.1007/s002220050170

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


Let K be a number field. Let W be a set of non-archimedean primes of K, let OK, W={xKordpx≥0∀pW}. 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 OK, W. (Thus, the Diophantine problem of OK, W is undecidable.)

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© Springer-Verlag Berlin Heidelberg 1997