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

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
Full Text: DOI
[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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