Characterizing integers among rational numbers with a universal-existential formula

Details Characterizing integers among rational numbers with a universal-existential formula Characterizing integers among rational numbers with a universal-existential formula

2007-03-30

arxiv.org/abs/math/0703907v1

Mathematics

Number Theory

6 pages

Scientific paper

We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defines the ring of integers.

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