Journal article
Defining ℤ in ℚ
- Abstract:
- We show that Z is definable in Q by a universal first-order formula in the language of rings. We also present an ∀∃-formula for Z in Q with just one universal quantifier. We exhibit new diophantine subsets of 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 non-squares is diophantine. Finally, we show that there is no existential formula for Z in Q, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over Q with many Q-rational points.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
Actions
Access Document
- Files:
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(Preview, Accepted manuscript, pdf, 228.5KB, Terms of use)
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- Publisher copy:
- 10.4007/annals.2016.183.1.2
Authors
- Publisher:
- Princeton University, Department of Mathematics
- Journal:
- Annals of Mathematics More from this journal
- Volume:
- 183
- Issue:
- 1
- Pages:
- 73-93
- Publication date:
- 2016-01-01
- DOI:
- EISSN:
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1939-8980
- ISSN:
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0003-486X
- Keywords:
- Pubs id:
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pubs:146876
- UUID:
-
uuid:bf44cb00-150e-4e4d-80c5-a123ae5cc21d
- Local pid:
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pubs:146876
- Source identifiers:
-
146876
- Deposit date:
-
2017-09-14
Terms of use
- Copyright holder:
- Annals of Mathematics
- Copyright date:
- 2016
- Notes:
- Annals of Mathematics © 2016. This is the accepted manuscript version of the article. The final version is available online from Princeton University, Department of Mathematics at: https://doi.org/10.4007/annals.2016.183.1.2
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