Defining $\mathbb{Z}$ in $\mathbb{Q}$

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 of the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for $\mathbb{Z}$ in $\mathbb{Q}$, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over $\mathbb{Q}$ with many $\mathbb{Q}$-rational points.

New shorter proofs of Proposition 16 (a) and Corollary 23, improved presentation and many tiny corrections