Characterizing integers among rational numbers with a universal-existential formula

• Autores: Bjorn Poonen
• Localización: American journal of mathematics, ISSN 0002-9327, Vol. 131, Nº 3, 2009, págs. 675-682
• Idioma: inglés
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• Resumen

  • 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.