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

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.)