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Hilbert's tenth problem for a class of rings of algebraic integers


Author: Thanases Pheidas
Journal: Proc. Amer. Math. Soc. 104 (1988), 611-620
MSC: Primary 12L05; Secondary 03C60, 03D35, 11U05
DOI: https://doi.org/10.1090/S0002-9939-1988-0962837-4
MathSciNet review: 962837
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Abstract: We show that $ {\mathbf{Z}}$ is diophantine over the ring of algebraic integers in any number field with exactly two nonreal embeddings into $ {\mathbf{C}}$ of degree $ \geq 3$ over $ {\mathbf{Q}}$.


References [Enhancements On Off] (What's this?)

  • [1] Z. I. Borevich and I. R. Shafarevich, Number theory, "Nauka", Moskow, 1964; English transl., Pure Appl. Math. 20, Academic Press, New York 1966. MR 0195803 (33:4001)
  • [2] J. Denef, Hilbert's Tenth Problem for quadratic rings, Proc. Amer. Math. Soc. 48 (1975), 214-220. MR 0360513 (50:12961)
  • [3] J. Denef, Diophantine sets over algebraic integer rings. II, Trans. Amer. Math. Soc. 257 (1980). MR 549163 (81b:12031)
  • [4] J. Denef and L. Lipshitz, Diophantine sets over some rings of algebraic integers, J. London Math. Soc. (2) 18 (1978), 385-391. MR 518221 (80a:12030)
  • [5] G. Hardy and E. Wright, An introduction to the theory of numbers, Oxford Univ. Press, Oxford, 1960.
  • [6] Yu. Matijasevič, Enumerable sets are diophantine, Dokl. Akad. Nauk SSSR 191 (1970), 272-282; English transl., Soviet Math. Dokl. 11 (1970), 354-357.

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DOI: https://doi.org/10.1090/S0002-9939-1988-0962837-4
Article copyright: © Copyright 1988 American Mathematical Society

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