Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Diophantine sets over algebraic integer rings. II

Author: J. Denef
Journal: Trans. Amer. Math. Soc. 257 (1980), 227-236
MSC: Primary 12L05; Secondary 10N05
MathSciNet review: 549163
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that Z is diophantine over the ring of algebraic integers in any totally real number field or quadratic extension of a totally real number field.

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

  • [1] Z. I. Borevich and I. R. Shafarevich, Number theory, ``Nauka", Moscow 1964; English transl., Pure and Appl. Math., vol. 20, Academic Press, New York, 1966. MR 0195803 (33:4001)
  • [2] M. Davis, Hilbert's tenth problem is unsolvable, Amer. Math. Monthly 80 (1973), 233-269. MR 0317916 (47:6465)
  • [3] M. Davis, Yu. Matijasevič and J. Robinson, Hilbert's tenth problem. Diophantine equations: positive aspects of a negative solution, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R. I., 1976, pp. 323-378. MR 0432534 (55:5522)
  • [4] J. Denef, Hilbert's tenth problem for quadratic rings, Proc. Amer. Math. Soc. 48 (1975), 214-220. MR 0360513 (50:12961)
  • [5] -, Diophantische verzamelingen over ringen van algebraische gehelen, Thesis, Leuven, 1976.
  • [6] 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)
  • [7] G. Hardy and E. Wright, An introduction to the theory of numbers, Oxford Univ. Press, Oxford, 1960.
  • [8] Yu. Matijasevič, Enumerable sets are diophantine, Dokl. Akad. Nauk SSSR 191 (1970), 279-282 (Russian); improved English translation: Soviet Math. Dokl. 11 (1970), 354-357.
  • [9] B. Mazur, Rational points on abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183-266. MR 0444670 (56:3020)
  • [10] O. T. O'Meara, Introduction to quadratic forms, 2nd ed., Springer-Verlag, Berlin, New York, 1971. MR 0347768 (50:269)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 12L05, 10N05

Retrieve articles in all journals with MSC: 12L05, 10N05

Additional Information

Keywords: Hilbert's tenth problem, unsolvable problems, diophantine equations
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society