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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

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