Journal Article
Defining ℤ in ℚ
Jochen Koenigsmann
Annals of Mathematics
SECOND SERIES, Vol. 183, No. 1 (January, 2016), pp. 73-93
Published
by: Mathematics Department, Princeton University
https://www.jstor.org/stable/24735167
Page Count: 21
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Topics: Diophantine sets, Integers, Mathematical rings, Mathematical theorems, Polynomials, Approximation, First order theories, Logical theorems
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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 nonsquares is diophantine.
Annals of Mathematics
© 2016 Annals of Mathematics
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