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Quadratic forms in normal open induction

Published online by Cambridge University Press:  12 March 2014

Margarita Otero
Margarita Otero*
Affiliation:
Departmento De Matematicas, Universidad Autónoma De Madrid, 28049 Madrid, Spain, E-mail: otero@ccuam3.sdi.uam.es
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Abstract

Models of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings.

Here we study the problem of representability of an element a of a model M of NOI (in some extension of M) by a quadratic form of the type X2 + b Y2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove that except in some trivial cases, M[x, y] with x2 + by2 = a can be embedded in a model of NOI.

We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M. Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 2 , June 1993 , pp. 456 - 476
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Adamowicz, Z., Some results on open and Diophantine induction, Logic Colloquium '84 (Paris, J. B.et al., editors), Elsevier, Amsterdam, 1986, pp. 1–20.Google Scholar
[2]Borevich, Z. and Shafarevich, I., Number Theory, Academic Press, New York, 1966.Google Scholar
[3]Bourbaki, N., Éléments de Mathématique. Algèbre Commutative, Hermann, Paris 1964, Chapters 5 and 6.Google Scholar
[4]Dries, L. V. D., Some model theory and number theory for models of weak systems of arithmetic (Pacholski, L.et al., editors), Model theory of algebra and arithmetic, Lecture Notes in Mathematics, vol. 834, Springer Verlag, Berlin, 1980, pp. 346–362.CrossRefGoogle Scholar
[5]Hardy, G. and Wright, E., An introduction to the theory of numbers, Oxford University Press, London and New York, 1979.Google Scholar
[6]Macintyre, A., and Marker, D., Primes and their residues rings in models of open induction, Annals of Pure and Applied Logic, vol. 43 (1989), pp. 57–77.CrossRefGoogle Scholar
[7]Nagata, M., A general theory of algebraic geometry over Dedekind Domains II, American Journal of Mathematics, vol. 80 (1958), pp. 382–420.CrossRefGoogle Scholar
[8]Otero, M., On Diophantine equations solvable in models of open induction, this Journal, vol. 55 (1990), pp. 779–786.Google Scholar
[9]Otero, M., Models of open induction, D. Phil Thesis., University of Oxford, 1991.Google Scholar
[10]Serre, J.-P., A course in arithmetic, Springer-Verlag, Berlin and New York, 1973.CrossRefGoogle Scholar
[11]Shepherdson, J., A nonstandard model for a free variable fragment of number theory, Bulletin of the Polish Academy of Sciences, vol. 12 (1964), pp. 79–86.Google Scholar
[12]Wilkie, A., Some results and problems on weak systems of arithmetic (Macintyre, A.et al., editors), Logic Colloquium '77, North-Holland, Amsterdam, 1978, pp. 285–296.CrossRefGoogle Scholar