Diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2

Journal of Number Theory

Volumn 95, Issue 2, 2002, Pages 227-252

  Shlapentokh, Alexandra ^a  

^a EAST CAROLINA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036695944     PISSN: 0022314X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-314X(01)92759-3     Document Type: Article

References (17)

1. J.-L. Colliot-Thélène, A. N. Skorobogatov, and P. Swinnerton-Dyer, Double fibres and double covers: Paucity of rational points, Acta Arith. 79 (1997), 113-135.
2. M. Davis, Hilbert's Tenth Problem is unsolvable, Amer. Math. Monthly 80 (1973), 233-269.
3. M. Davis, Yu. Matijasevich, and J. Robinson, Positive aspects of a negative solution, Proc. Symp. Pure Math., Amer. Math. Soc. 28 (1976), 323-378.
4. J. Denef, Hilbert's Tenth Problem for quadratic rings, Proc. Amer. Math. Soc. 48 (1975), 214-220.
5. J. Denef, and L. Lipschitz, Diophantine sets over some rings of algebraic integers, J. London Math. Soc. (2) 18 (1978), 385-391.
6. J. Denef, Diophantine sets of algebraic integers. II, Trans. Amer. Math. Soc. 257 (1), (1980), 227-236.
7. M. Fried and M. Jarden, "Field Arithmetic," Springer-Verlag, New York, 1986.
8. G. Januz, "Algebraic Number Fields," Academic Press, New York, 1973.
9. S. Lang, "Algebraic Number Theory," Addison-Wesley, Reading, MA, 1970.
10. B. Mazur, The topology of rational points, Experiment. Math. 1 (1992), 35-45.
11. B. Mazur, Questions of decidability and undecidability in number theory, J. Symbolic Logic 59 (1994), 353-371.
12. T. Pheidas, Hilbert's Tenth Problem for a class of rings of algebraic integers, Proc. Amer. Math. Soc. 104 (1988), 611-620.
13. H. N. Shapiro and A. Shlapentokh, Diophantine relations between algebraic number fields, Comm. Pure Appl. Math. XLII (1989), 1113-1122.
14. A. Shlapentokh, Extension of Hilbert's Tenth Problem to some algebraic number fields, Comm. Pure Appl. Math. XLII (1989), 939-962.
15. A. Shlapentokh, Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator, Invent. Math. 129 (1997), 489-507.
16. A. Shlapentokh, Diophantine classes of holomorphy rings of global fields, J. Algebra 169 (1994), 139-175.
17. A. Shlapentokh, Defining integrality at prime sets of high density in number fields, Duke Math. J. 101 (2000), 117-134.