Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T14:54:03.116Z Has data issue: false hasContentIssue false
  • English
  • Français

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 (jonathan.kirby@uea.ac.uk)
Angus Macintyre
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK (a.macintyre@qmul.ac.uk)
Alf Onshuus
Affiliation:
Departamento de Matemáticas, Universidad de los Andes, Kr 1 No. 18A-10, Edificio H, Bogotá, Colombia (aonshuus@uniandes.edu.co)
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Gonshor, H., An introduction to the theory of surreal numbers, London Mathematical Society Lecture Note Series, Volume 110 (Cambridge University Press, 1986).CrossRefGoogle Scholar
2.Jacobson, N., Lectures in abstract algebra, III: Theory of fields and Galois theory, Graduate Texts in Mathematics, Volume 32 (Springer, 1975).Google Scholar
3.Kirby, J., Finitely presented exponential fields, preprint (arXiv:0912.4019v4; 2011).Google Scholar
4.Kirby, J., Exponential algebraicity in exponential fields, Bull. Lond. Math. Soc. 42(5) (2010), 879–890.CrossRefGoogle Scholar
5.Kirby, J., A note on the axioms for Zilber's pseudo-exponential fields, preprint (arXiv: 1006.0894v2; 2011).Google Scholar
6.Kirby, J., On quasiminimal excellent classes, J. Symb. Logic 75(2) (2010), 551–564.CrossRefGoogle Scholar
7.Laczkovich, M., The removal of π from some undecidable problems involving elementary functions, Proc. Am. Math. Soc. 131(7) (2003), 2235–2240.CrossRefGoogle Scholar
8.Macintyre, A., Surreal numbers, unpublished notes (1999).Google Scholar
9.Macintyre, A., The complex exponential field is not model complete, unpublished notes (1991).Google Scholar
10.Macintyre, A. and Wilkie, A. J., On the decidability of the real exponential field, in Kreiseliana, pp. 441–467 (A. K. Peters, Wellesley, MA, 1996).Google Scholar
11.Mantova, V., Involutions on Zilber fields, preprint (arXiv:1109.6155; 2011).CrossRefGoogle Scholar
12.Marker, D., A remark on Zilber's pseudoexponentiation, J. Symb. Logic 71(3) (2006), 791–798.CrossRefGoogle Scholar
13.Tarski, A., A decision method for elementary algebra and geometry, 2nd edn (University of California Press, Berkeley/Los Angeles, CA, 1951).CrossRefGoogle Scholar
14.van den Dries, L. and Ehrlich, P., Fields of surreal numbers and exponentiation, Fund. Math. 167(2) (2001), 173–188.CrossRefGoogle Scholar
15.van den Dries, L., Macintyre, A. and Marker, D., Logarithmic-exponential power series, J. Lond. Math. Soc. 56(3) (1997), 417–434.CrossRefGoogle Scholar
16.Wilkie, A. J., Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Am. Math. Soc. 9(4) (1996), 1051–1094.CrossRefGoogle Scholar
17.Zilber, B., Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals Pure Appl. Logic 132(1) (2005), 67–95.CrossRefGoogle Scholar