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Transactions of the American Mathematical Society

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Undecidable existential problems for addition and divisibility in algebraic number rings


Author: L. Lipshitz
Journal: Trans. Amer. Math. Soc. 241 (1978), 121-128
MSC: Primary 02E10; Secondary 02G05, 10N10
DOI: https://doi.org/10.1090/S0002-9947-1978-0536658-9
MathSciNet review: 0536658
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Abstract: Existential formulas involving addition and divisibility are shown to be undecidable in the ring of integers of a real quadratic extension of the rationals. A weaker result is proved for extensions of higher degree.


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

  • [1] A. P. Bel'tyukov, Abstracts: Students' Scientific Conference, Leningrad University, 1975.
  • [2] Z. I. Borevich and I. R. Shafarevich, Number theory, Academic Press, New York, 1966. MR 33 #4001. MR 0195803 (33:4001)
  • [3] R. D. Carmichael, On the numerical factors of the arithmetical forms $ {\alpha ^n} + {\beta ^n}$, Ann. of Math. (2) 15 (1913), 30-70. MR 1502458
  • [4] J. Denef, Hilbert's tenth problem for quadratic rings, Proc. Amer. Math. Soc. 48 (1975), 214-220. MR 0360513 (50:12961)
  • [5] L. Lipshitz, The Diophantine problem for addition and divisibility, Trans. Amer. Math. Soc. (to appear). MR 0469886 (57:9666)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0536658-9
Article copyright: © Copyright 1978 American Mathematical Society

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