Geodesic
Defining ${\mathbb Z}$ in ${\mathbb Q}$
Jochen Koenigsmann1
1 Mathematical Institute, University of Oxford, Oxford, UK
Annals of mathematics, Tome 183 (2016) no. 1, pp. 73-93.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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.
DOI : 10.4007/annals.2016.183.1.2

Jochen Koenigsmann 1

1 Mathematical Institute, University of Oxford, Oxford, UK 
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Jochen Koenigsmann. Defining ${\mathbb Z}$ in ${\mathbb Q}$. Annals of mathematics, Tome 183 (2016) no. 1, pp. 73-93. doi : 10.4007/annals.2016.183.1.2. http://geodesic.mathdoc.fr/articles/10.4007/annals.2016.183.1.2/

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