Characterizing integers among rational numbers with a universal-existential formula

By Bjorn Poonen. 2007

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We prove that ${\Bbb Z}$ in definable in ${\Bbb Q}$ by a formula with two universal quantifiers followed by seven existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of ${\Bbb Q}$-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.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/16d130e0225fff41cce757396b0e5f2f0ce0d301