proceedings of the american mathematical society Volume 104, Number 2, October 1988 HILBERT'S TENTH PROBLEM FOR A CLASS OF RINGS OF ALGEBRAIC INTEGERS THANASES PHEIDAS (Communicated by Thomas J. Jech) ABSTRACT. We show that Z is diophantine over the ring of algebraic integers in any number field with exactly two nonreal embeddings into C of degree > 3 over Q. Introduction. Let R be a ring. A set S C Rm is called diophantine over R if it is of the form S = {x G Rm : 3t/ G Rn p(x, y) = 0}, where p is a polynomial in R[x,y). A number field is a finite extension of the field Q of rational numbers. If K is a number field, we denote by Ok the ring of elements of K which are integral over the ring Z of rational integers. N is the set {0,1,2,...} and N0 is the set {1,2,3,... }. In this paper we prove THEOREM. Let K be a number field of degree n > 3 over Q with exactly two nonreal embeddings into the field C of complex numbers. Then Z is diophantine over 0K. An example of such a number field is Q(d) where c/3 is a rational number which does not have a rational cube root. In order to prove the theorem, we use the methods of J. Denef in [3]. The terminology and enumeration of the lemmas is kept the same as in [3] so that the similarities and differences of the proofs are clear. The theorem implies COROLLARY. Let K be as in the theorem. Then Hilbert's Tenth Problem in Ok is undecidable. The results of [3] and the present paper are the maximum that can be achieved using the present methods. Hence the general conjecture made in [4], namely that Hilbert's Tenth Problem for the integers of any number field is undecidable, remains open. Let if be a number field of degree n > 3 over Q with exactly two nonreal embeddings into C. Let <t¿, i = 1,2, ...,n, be all the embeddings of K into C, enumerated in such a way that crn_i and an are nonreal. Then the embedding a: K —► C such that a(x) — an(x) is distinct from an and from all ai, i < n — 2, Received by the editors July 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 03B25; Secondary 12B99. I would like to thank Professor Leonard Lipschitz for his encouragement and help during the preparation of this work. In the process of publication of this paper I was informed that Alexandra Shlapentokh obtained the same results as part of her thesis at Courant Institute of Mathematical Sciences. This paper has been supported in part by NSF Grant #DMS 8605-198. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page 611 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use