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

摘要

Let K be a number field. Let W be a set of non-archimedean primes of K, let OK, W={x∈Kordpx≥0pW}. Then if K is a totally real non-trivial cyclic extension of á, there exists an infinite set W of finite primes of K such that ê and the ring of algebraic integers of K have a Diophantine definition over OK, W. (Thus, the Diophantine problem of OK, W is undecidable.)