Abstract
We show that ${\mathbb Z}$ is definable in ${\mathbb Q}$ by a universal first-order formula in the language of rings. We also present an $\forall\exists$-formula for ${\mathbb Z}$ in ${\mathbb Q}$ with just one universal quantifier. We exhibit new diophantine subsets of ${\mathbb Q}$ like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof for the fact that the set of nonsquares is diophantine.
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@article{CCS, mrkey = {0592151},
author = {Colliot-Th{é}l{è}ne, Jean-Louis and Coray, Daniel and Sansuc, Jean-Jacques},
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coden = {JRMAA8},
mrclass = {14G05 (10C02)},
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