Defining Z in Q

Details Defining Z in Q Defining Z in Q

2010-11-15

arxiv.org/abs/1011.3424v1

Mathematics

Number Theory

Scientific paper

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 set of non-squares or the complement of the image of the norm map under a quadratic extension. 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.

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