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The algebraic numbers definable in various exponential fields

Part of: Number theory: Connections with logic Model theory

Published online by Cambridge University Press:  02 April 2012

Jonathan Kirby ,
Angus Macintyre  and
Alf Onshuus
Jonathan Kirby
Affiliation:
School of Mathematics, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, UK ([email protected])
Angus Macintyre
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK ([email protected])
Alf Onshuus
Affiliation:
Departamento de Matemáticas, Universidad de los Andes, Kr 1 No. 18A-10, Edificio H, Bogotá, Colombia ([email protected])
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Abstract

We prove the following theorems. Theorem 1: for any E-field with cyclic kernel, in particular ℂ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: for the Zilber fields, the only pointwise definable algebraic numbers are the real abelian numbers.

Keywords

Zilber modelsSchanuel conjecturecomplex exponential field

MSC classification

Secondary: 03C35: Categoricity and completeness of theories 11U09: Model theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 11 , Issue 4 , October 2012 , pp. 825 - 834
Copyright
Copyright © Cambridge University Press 2012

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