Inventiones mathematicae

, Volume 129, Issue 3, pp 489-507

First online:

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

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

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Let K be a number field. Let W be a set of non-archimedean primes of K, let O K , W ={xKord p x≥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 O K , W . (Thus, the Diophantine problem of O K , W is undecidable.)