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Die böse Farbe

Part of: Model theory

Published online by Cambridge University Press:  11 December 2007

Andreas Baudisch ,
Martin Hils ,
Amador Martin-Pizarro  and
Frank O. Wagner
Andreas Baudisch
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany (baudisch@mathematik.hu-berlin.de; hils@mathematik.hu-berlin.de).
Martin Hils
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany (baudisch@mathematik.hu-berlin.de; hils@mathematik.hu-berlin.de). Equipe de Logique Mathématique, Université Denis-Diderot Paris VII, 75251 Paris Cedex 05, France. Université de Lyon; Université Lyon 1, CNRS, Institut Camille Jordan, 43 bd du 11 nov. 1918, 69622 Villeurbanne Cedex, France, (pizarro@math.univ-lyon1.fr; wagner@math.univ-lyon1.fr).
Amador Martin-Pizarro
Affiliation:
Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany (baudisch@mathematik.hu-berlin.de; hils@mathematik.hu-berlin.de). Université de Lyon; Université Lyon 1, CNRS, Institut Camille Jordan, 43 bd du 11 nov. 1918, 69622 Villeurbanne Cedex, France, (pizarro@math.univ-lyon1.fr; wagner@math.univ-lyon1.fr).
Frank O. Wagner
Affiliation:
Université de Lyon; Université Lyon 1, CNRS, Institut Camille Jordan, 43 bd du 11 nov. 1918, 69622 Villeurbanne Cedex, France, (pizarro@math.univ-lyon1.fr; wagner@math.univ-lyon1.fr).
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Abstract

We construct a bad field in characteristic zero. That is, we construct an algebraically closed field which carries a notion of dimension analogous to Zariski-dimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a longstanding open question by Zilber.

Zusammenfassung

Wir konstruieren einen schlechten Körper der Charakteristik Null. Mit anderen Worten, wir konstruieren einen algebraisch abgeschlossenen Körper mit einem Dimensionsbegriff analog der Zariski-Dimension, zusammen mit einer unendlichen echten multiplikativen Untergruppe der Dimension Eins, so daβ der Körper selbst Dimension Zwei hat. Dies beantwortet eine alte Frage von Zilber.

Keywords

ModelltheorieKörper von endlichem MorleyrangSchlechter KörperAmalgamierungskonstruktionmodel theoryfields of finite Morley rankbad fieldamalgamation construction

MSC classification

Secondary: 03C65: Models of other mathematical theories 03C50: Models with special properties (saturated, rigid, etc.) 03C45: Classification theory, stability and related concepts 03C35: Categoricity and completeness of theories
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 3 , July 2009 , pp. 415 - 443
Copyright
Copyright © Cambridge University Press 2009

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References

Literatur

1.Ax, J., On Schanuel's conjecture, Annals Math. 93 (1971), 252–258.CrossRefGoogle Scholar
2.Baldwin, J. und Holland, K., Constructing ω-stable structures: fields of rank 2, J. Symb. Logic 65(1) (2000), 371–391.Google Scholar
3.Baudisch, A., Martin-Pizarro, A. und Ziegler, M., On fields and colours, Alg. Logika 45(2) (2006), 159–184.Google Scholar
4.Baudisch, A., Martin-Pizarro, A. und Ziegler, M., Hrushovski's fusion, in Festschrift fuer Ulrich Felgner zum 65 Geburtstag (Herausgeber F. Haug, B. Loewe und T. Schatz), Studies in Logic, Volume 4, pp. 15–31 (College Publications, London, 2007).Google Scholar
5.Baudisch, A., Martin-Pizarro, A. und Ziegler, M., Fusion over a vector space, J. Math. Logic 6(2) (2007), 141–162.Google Scholar
6.Baudisch, A., Martin-Pizarro, A. und Ziegler, M., Red fields, J. Symb. Logic 72(1) (2007), 207–225.CrossRefGoogle Scholar
7.Goode, J. B., Hrushovski's geometries, Seminarberichte der Humboldt Universität zu Berlin, Nr. 104, pp. 106–117 (Humboldt Universität zu Berlin, 1989).Google Scholar
8.Hasson, A. und Hils, M., Fusion over sublanguages, J. Symb. Logic 71(2) (2006), 361–398.CrossRefGoogle Scholar
9.Hrushovski, E., Strongly minimal expansions of algebraically closed fields, Israel J. Math. 79 (1992), 129–151.CrossRefGoogle Scholar
10.Hrushovski, E., A new strongly minimal set, Annals Pure Appl. Logic 62 (1993), 147–166.Google Scholar
11.Kirby, J., The theory of the exponential differential equations of semiabelian varieties, preprint arXiv:0708.1352v1 (2007).Google Scholar
12.Macintyre, A., On ω1-categorical fields, Fund. Math. 71 (1971), 1–25.CrossRefGoogle Scholar
13.Mustafin, E., Structure des groupes linéaires définissables dans un corps de rang de Morley fini, J. Alg. 281(2) (2004), 753–773.Google Scholar
14.Poizat, B., Le carré de l'égalité, J. Symb. Logic 64(3) (1999), 1338–1355.Google Scholar
15.Poizat, B., L'égalité au cube, J. Symb. Logic 66(4) (2001), 1647–1676.Google Scholar
16.van den Dries, L. und Schmidt, K., Bounds in the theory of polynomial rings over fields, Invent. Math. 76 (1984), 77–91.CrossRefGoogle Scholar
17.Wagner, F. O., Fields of finite Morley rank, J. Symb. Logic 66(2) (2001), 703–706.CrossRefGoogle Scholar
18.Wagner, F. O., Bad fields in positive characteristic, Bull. Lond. Math. Soc. 35 (2003), 499–502.Google Scholar
19.Ziegler, M., Lemma für Daniels beschränkte Automorphismen (‘A note on generic types’), preprint arXiv math/0608433v1 (2004).Google Scholar
20.Zilber, B. I., Groups and rings whose theory is categorical, Fund. Math. 95 (1977), 173–188.Google Scholar