Advertisement

Israel Journal of Mathematics

, Volume 86, Issue 1–3, pp 25–59 | Cite as

Pseudo algebraically closed fields over rings

  • Moshe Jarden
  • Aharon Razon
Article

Abstract

We prove that for almost allσG ℚ the field\(\tilde{\mathbb{Q}}\) has the following property: For each absolutely irreducible affine varietyV of dimensionr and each dominating separable rational mapϕ:V\(\tilde{\mathbb{Q}}\) there exists a point a ∈\(\mathbb{A}\) such thatϕ(a) ∈ ℤr. We then say that\(\tilde{\mathbb{Q}}\) is PAC over ℤ. This is a stronger property then being PAC. Indeed we show that beside the fields\(\tilde{\mathbb{Q}}\) other fields which are algebraic over ℤ and are known in the literature to be PAC are not PAC over ℤ.

Keywords

Finite Group Branch Point Galois Extension Irreducible Polynomial Finite Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ax]
    J. Ax,The elementary theory of finite fields, Annals of Mathematics88 (1968), 239–271.CrossRefMathSciNetGoogle Scholar
  2. [CaF]
    J.W.S. Cassels and A. Fröhlich,Algebraic Number Theory, Academic Press, London, 1967.MATHGoogle Scholar
  3. [Fal]
    G. Faltings,Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inventiones Mathematicae73 (1983), 349–366.CrossRefMathSciNetMATHGoogle Scholar
  4. [Fre]
    G. Frey,Pseudo algebraically closed fields with non-archimedian real valuations, Journal of Algebra26 (1973), 202–207.MATHCrossRefMathSciNetGoogle Scholar
  5. [FJ1]
    M. Fried and M. Jarden, Diophantine properties of subfields of\(\tilde{\mathbb{Q}}\), American Journal of Mathematics100 (1978), 653–666.MATHCrossRefMathSciNetGoogle Scholar
  6. [FJ2]
    M. D. Fried and M. Jarden,Field Arithmetic, Ergebnisse der Mathematik (3)11, Springer, Heidelberg, 1986.MATHGoogle Scholar
  7. [FrV]
    M. Fried and H. Völklein,The inverse Galois problem and rational points on moduli spaces, Mathematische Annalen290 (1991), 771–800.MATHCrossRefMathSciNetGoogle Scholar
  8. [GaJ]
    W.-D. Geyer and M. Jarden,On stable fields in positive characteristic, Geometria Dedicata29 (1989), 335–375.MATHCrossRefMathSciNetGoogle Scholar
  9. [HJ1]
    D. Haran and M. Jarden,The absolute Galois group of a pseudo p-adically closed field, Journal für die reine und angewandte Mathematik383 (1988), 147–206.MATHMathSciNetCrossRefGoogle Scholar
  10. [HJ2]
    D. Haran and M. Jarden,The absolute Galois group of a pseudo real closed field, Annali della Scuola Normale Superiore — Pisa, Serie IV,12 (1985), 449–489.MATHMathSciNetGoogle Scholar
  11. [Har]
    D. Harbater,Galois coverings of the arithmetic line, Lecture Notes in Mathematics1240, Springer, Berlin, 1987, pp. 165–195.Google Scholar
  12. [Jar]
    M. Jarden,Elementary statements over large algebraic fields, Transactions of AMS164 (1972), 67–91.MATHCrossRefMathSciNetGoogle Scholar
  13. [Ja2]
    M. Jarden,Intersection of local algebraic extensions of a Hilbertian field (edited by A. Barlotti et al.), NATO ASI Series C333, Kluwer, Dordrecht, 1991, pp. 343–405.Google Scholar
  14. [Ja3]
    M. Jarden,The inverse Galois problem over formal power series fields, Israel Journal of Mathematics85 (1994), 263–275.MATHCrossRefMathSciNetGoogle Scholar
  15. [Ja]
    M. Jarden and Peter Roquette,The Nullstellensatz over p-adically closed fields, Journal of the Mathematical Society of Japan32 (1980), 425–460.MATHMathSciNetCrossRefGoogle Scholar
  16. [La1]
    S. Lang,Introduction to Algebraic Geometry, Interscience Publishers, New York, 1958.MATHGoogle Scholar
  17. [La2]
    S. Lang,Algebra, Addison-Wesley, Reading, 1970.MATHGoogle Scholar
  18. [La3]
    S. Lang,Algebraic Number Theory, Addison-Wesley, Reading, 1970.MATHGoogle Scholar
  19. [Mat]
    B. H. Matzat,Der Kenntnisstand in der Konstruktiven Galoischen Theorie, manuscript, Heidelberg, 1990.Google Scholar
  20. [Mum]
    D. Mumford,The Red Book of Varieties and Schemes, Lecture Notes in Mathematics1358, Springer, Berlin, 1988.MATHGoogle Scholar
  21. [Pop]
    F. Pop,Fields of totally Σ-adic numbers, manuscript, Heidelberg, 1992.Google Scholar
  22. [Pre]
    A. Prestel,Pseudo real closed fields, inSet Theory and Model Theory, Lecture Notes in Math.872, Springer, Berlin, 1981, pp. 127–156.CrossRefGoogle Scholar
  23. [RCVS]
    M. Rzedowski-Calderón and G. Villa-Salvador,Automorphisms of congruence function fields, Pacific Journal of Mathematics150 (1991), 167–178.MATHMathSciNetGoogle Scholar
  24. [Sam]
    P. Samuel,Lectures on Old and New Results on Algebraic Curves, Tata Institute of Fundamental Research, Bombay, 1966.MATHGoogle Scholar
  25. [Se1]
    J.-P. Serre,Topics in Galois Theory, Jones and Barlett, Boston, 1992.MATHGoogle Scholar
  26. [Se2]
    J.-P. Serre,A Course in Arithmetic, Graduate Texts in Mathematics7, Springer, New York, 1973.MATHGoogle Scholar
  27. [Sha]
    I.R. Shafarevich,Basic algebraic Geometry, Grundlehren der mathematischen Wissenschaften213, Springer, Berlin, 1977.MATHGoogle Scholar
  28. [Voe]
    H. Völklein, Braid groups, Galois groups and cyclic covers of\(\mathbb{P}\), manuscript, 1992.Google Scholar

Copyright information

© The Magnes Press 1994

Authors and Affiliations

  1. 1.School of Mathematical SciencesRaymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv UniversityTel AvivIsrael

Personalised recommendations