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

Author: Shlapentokh A.

Source: Journal of Number Theory, Volume 95, Number 2, August 2002 , pp. 227-252(26)

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Abstract:

Let M be a number field. Let W be a set of non-archimedean primes of M. LetO_M,W={xM mid ord_px ges 0 forall p notin W}.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 extension of degree 2 of a totally real field. Then, for any >0, there exists a set W_M of primes of M whose density is greater than 1-[M: Qopf ]^-1- epsiv and such that Zopf has a Diophantine definition over O_{M,W_M}. (Thus, Hilbert's Tenth Problem is undecidable in O_{M,W_M}.) (2) Let M be as above and let >0 be given. Let S be the set of all rational primes splitting in M. (If the extension is Galois but not cyclic, S contains all the rational primes.) Then there exists a set of M-primes W_M such that the set of rational primes W below W_M differs from S by a set contained in a set of density less than and such that Zopf has a Diophantine definition over O_{M,W_M}. (Again this will imply that Hilbert's Tenth Problem is undecidable in O_{M,W_M}.) © 2002 Elsevier Science (USA).