Skip to main content
Log in

A Note on Complex p-Adic Exponential Fields

  • Research Articles
  • Published:
p-Adic Numbers, Ultrametric Analysis and Applications Aims and scope Submit manuscript

Abstract

In this paper we apply Ax-Schanuel’s Theorem to the ultraproduct of p-adic fields in order to get some results towards algebraic independence of p-adic exponentials for almost all primes p.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. J. Ax, “On Schanuel’s conjectures,” Annals Math., Second Ser. 93 (2), 252–268 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. J. Engler and A. Prestel, Valued Fields, SpringerMonographs in Mathematics (2005).

    MATH  Google Scholar 

  3. R. Bianconi, “Some remarks on Schanuel’s conjecture,” Annals Pure Appl. Logic 108, 15–18 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  4. C. C. Chang and H. J. Keisler, Model Theory (North-Holland Publ. Co., Amsterdam, 1973).

    Book  MATH  Google Scholar 

  5. K. Gravett, “Ordered Abelian groups,” Quarterly J.Math., Oxford Second Ser. 7, 57–63 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Hahn, “Über die nichtarchimedischen Grössensystem,” Sitzungsberichte der Kaiserlichen Akademie derWissenschaften,Mathematisch-Naturwissenschaftliche Klasse (Wien) 116, no. Abteilung IIa, 601–655 (1907).

    MATH  Google Scholar 

  7. J. van der Hoeven, “Operators on generalized power series,” Illinois J. Math. 45, 1161–1190 (2001).

    MathSciNet  MATH  Google Scholar 

  8. I. Kaplansky, “Maximal fields with valuations,” Duke Math. J. 9, 303–321 (1942).

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Kuhlmann and M. Matusinski, “Hardy type derivations on fields of exponential logarithmic series,” J. Algebra 345, 171–189 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Kuhlmann, M. Matusinski and A. Shkop, “A note on Schanuel’s conjectures for exponential logarithmic power series fields,” A.C. Arch.Math. 100, 431 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Matusinski, “On generalized series fields and exponential-logarithmic series fields with derivations,” Second Int. Conference and Workshop on Valuation Theory, Segovia /El Escorial (Spain), Vol.: Valuation Theory in Interaction (EMS Series of Congress Reports), [arXiv:1209.4559v3].

  12. B. H. Neumann, “On ordered division rings,” Transact. Amer.Math. Soc. 66, 202–252 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  13. A.M. Robert, A Course in p-Adic Analysis, Graduate Texts inMathematics 198 (Springer-Verlag, 2000).

  14. H. Schoutens, The Use of Ultraproducts in Commutative Algebra, Lecture Notes in Mathematics (Springer-Verlag, Berlin, Heidelberg, 2010).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ali Bleybel.

Additional information

The text was submitted by the author in English.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bleybel, A. A Note on Complex p-Adic Exponential Fields. P-Adic Num Ultrametr Anal Appl 10, 267–275 (2018). https://doi.org/10.1134/S2070046618040039

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070046618040039

Key words

Navigation