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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