Diophantine Definability and Decidability in Large Subrings of Totally Real Number Fields and Their Totally Complex Extensions of Degree 2

A Shlapentokh
2002 Journal of Number Theory  
Let M be a number field. Let W be a set of non-archimedean primes of M. Let O M;W ¼ fx 2 M j ord p x50 8peW g: The author continues her investigation of Diophantine definability and decidability in rings O M;W where W is infinite. In this paper, she improves her previous density estimates and extends the results to the totally complex extensions of degree 2 of the totally real fields. In particular, the following results are proved: (1) Let M be a totally real field or a totally complex
more » ... n of degree 2 of a totally real field. Then, for any e > 0, there exists a set W M of primes of M whose density is greater than 1 À ½M : Q À1 À e and such that Z has a Diophantine definition over O M;WM . (Thus, Hilbert's Tenth Problem is undecidable in O M;WM .) (2) Let M be as above and let e > 0 be given. Let S Q be the set of all rational primes splitting in M. (If the extension is Galois but not cyclic, S Q contains all the rational primes.) Then there exists a set of M-primes W M such that the set of rational primes W Q below W M differs from S Q by a set contained in a set of density less than e and such that Z has a Diophantine definition over O M;WM . (Again this will imply that Hilbert's Tenth Problem is undecidable in O M;WM .) #
doi:10.1016/s0022-314x(01)92759-3 fatcat:nfau4nyfura25j7tsftc3xgeaa