Journal article

### 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 ...

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Institution:
University of Oxford
Department:
Oxford, MPLS, Mathematical Inst
Publication date:
2010-11-15
URN:
Source identifiers:
146876
Local pid:
pubs:146876
Keywords: