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Manuscripta Mathematica

, Volume 134, Issue 3–4, pp 533–544 | Cite as

Embeddings of function fields into ample fields

  • Arno FehmEmail author
Article
  • 69 Downloads

Abstract

Let E be an ample field and \({K\subseteq E}\) a subfield. We determine when a function field F/K can be embedded into E and compute the number of such embeddings. We give some applications and exhibit new examples of non-ample fields.

Mathematics Subject Classification (2000)

12E30 12F20 14H05 

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References

  1. 1.
    Azgin, S., Kuhlmann, F.V., Pop, F.: Characterization of extremal valued fields. Manuscript (2010)Google Scholar
  2. 2.
    Bogomolov F., Tschinkel Y.: Unramified correspondences. In: Vostokov, S., Zarhin, Yu. (eds) Algebraic Number Theory and Algebraic Geometry, Contemporary Mathematics, vol. 300, pp. 17–26. American Mathematical Society, Providence (2002)Google Scholar
  3. 3.
    Bourbaki N.: Commutative Algebra. Hermann, Paris (1972)zbMATHGoogle Scholar
  4. 4.
    Colliot-Thélène J.-L.: Rational connectedness and Galois covers of the projective line. Ann. Math. 151(1), 359–373 (2000)zbMATHCrossRefGoogle Scholar
  5. 5.
    Dèbes, P.: Galois covers with prescribed fibers: the Beckmann–Black problem. In: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome, vol. 2, no. 28, pp. 273–286 (1998)Google Scholar
  6. 6.
    Dèbes P., Deschamps B.: The inverse Galois problem over large fields. In: Schneps, L., Lochak, P. (eds) Geometric Galois Action, vol. 2, London Mathematical Society Lecture Note Series, vol. 243, pp. 119–138. Cambridge University Press, Cambridge (1999)Google Scholar
  7. 7.
    Fehm A.: Subfields of ample fields. Rational maps and definability. J. Algebra 323(6), 1738–1744 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fehm A., Petersen S.: On the rank of abelian varieties over ample fields. Int. J. Number Theory 6(3), 579–586 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fried M.D., Jarden M.: Field Arithmetic, 3rd edn. Springer, New York (2008)zbMATHGoogle Scholar
  10. 10.
    Garcia A., Stichtenoth H.: On the asymptotic behaviour of some towers of function fields over finite fields. J. Number Theory 61, 248–273 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Garcia A., Stichtenoth H., Thomas M.: On towers and composita of tower of function fields over finite fields. Finite Fields Appl. 3, 257–274 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Geyer W.-D., Jarden M.: Non-PAC fields whose henselian closures are separably closed. Math. Res. Lett. 8, 509–519 (2001)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Haran D., Jarden M.: Regular split embedding problems over function fields of one variable over ample fields. J. Algebra 208, 147–164 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Harbater D.: On function fields with free absolute Galois groups. J. für die reine und angewandte Mathematik 632, 85–103 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Jarden M.: On ample fields. Archiv der Mathematik 80, 475–477 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Junker M., Koenigsmann J.: Schlanke Körper (Slim Fields). J. Symbolic Logic 75(2), 481–500 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Koenigsmann J.: Defining transcendentals in function fields. J. Symbolic Logic 67(3), 947–956 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Koenigsmann J.: The regular inverse Galois problem over non-large fields. J. Europ. Math. Soc 6(4), 425–434 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kollár J.: Diophantine subsets of function fields of curves. Algebra Number Theory 2(3), 299–311 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kuhlmann, F.-V.: On local uniformization in arbitrary characteristic. I. The Fields Institute Preprint Series, Toronto. arXiv:math/9903097v1 [math.AG] (1998)Google Scholar
  21. 21.
    Kuhlmann F.-V.: Places of algebraic function fields in arbitrary characteristic. Adv. Math. 188, 399–424 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lang S., Tate J.: On Chevalley’s proof of Luroth’s theorem. Proc. AMS 3, 621–624 (1952)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Poonen B.: Unramified covers of Galois covers of low genus curves. Math. Res. Lett. 12(4), 475–482 (2005)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Poonen B.: Gonality of modular curves in characteristic p. Math. Res. Lett. 14(4), 691–701 (2007)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Poonen B., Pop F.: First-order definitions in function fields over anti-mordellic fields. In: Chatzidakis, Z., Macpherson, D., Pillay, A., Wilkie, A. (eds) Model Theory with Applications to Algebra and Analysis, Cambridge University Press, Cambridge (2007)Google Scholar
  26. 26.
    Pop F.: Embedding problems over large fields. Ann. Math. 144, 1–34 (1996)zbMATHCrossRefGoogle Scholar
  27. 27.
    Pop F.: Henselian implies large. Ann. Math. 172, 2183–2195 (2010)CrossRefGoogle Scholar
  28. 28.
    Prestel A., Roquette P.: Formally p-adic Fields. Springer, New York (1984)Google Scholar
  29. 29.
    Samuel P.: Lectures on Old and New Results on Algebraic Curves. Tata Institute, Bombay (1966)zbMATHGoogle Scholar
  30. 30.
    Tamagawa, A.: Unramified Skolem problems and unramified arithmetic Bertini theorems in positive characteristic. Documenta Mathematica, Extra Volume: Kazuya Kato’s Fiftieth Birthday, pp. 789–831 (2003)Google Scholar
  31. 31.
    Tressl M.: The uniform companion for large differential fields of characteristic 0. Trans. Am. Math. Soc. 357(10), 3933–3951 (2005)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Universität Konstanz, Fachbereich Mathematik und StatistikKonstanzGermany

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