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Diophantine sets over algebraic integer rings. II. (English) Zbl 0426.12009

MSC:
11R80 Totally real fields
03D25 Recursively (computably) enumerable sets and degrees
12L99 Connections between field theory and logic
11U05 Decidability (number-theoretic aspects)
11D99 Diophantine equations
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[1] A. I. Borevich and I. R. Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. · Zbl 0145.04902
[2] Martin Davis, Hilbert’s tenth problem is unsolvable, Amer. Math. Monthly 80 (1973), 233 – 269. · Zbl 0277.02008 · doi:10.2307/2318447 · doi.org
[3] Martin Davis, Yuri Matijasevič, and Julia Robinson, Hilbert’s tenth problem: Diophantine equations: positive aspects of a negative solution, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 323 – 378. (loose erratum).
[4] J. Denef, Hilbert’s tenth problem for quadratic rings, Proc. Amer. Math. Soc. 48 (1975), 214 – 220. · Zbl 0324.02032
[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), no. 3, 385 – 391. · Zbl 0399.10049 · doi:10.1112/jlms/s2-18.3.385 · doi.org
[7] G. Hardy and E. Wright, An introduction to the theory of numbers, Oxford Univ. Press, Oxford, 1960. · Zbl 0086.25803
[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] Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183 – 266. · Zbl 0245.14015 · doi:10.1007/BF01389815 · doi.org
[10] O. T. O’Meara, Introduction to quadratic forms, Springer-Verlag, New York-Heidelberg, 1971. Second printing, corrected; Die Grundlehren der mathematischen Wissenschaften, Band 117. · Zbl 0207.05304
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