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.
Similar content being viewed by others
References
J. Ax, “On Schanuel’s conjectures,” Annals Math., Second Ser. 93 (2), 252–268 (1971).
A. J. Engler and A. Prestel, Valued Fields, SpringerMonographs in Mathematics (2005).
R. Bianconi, “Some remarks on Schanuel’s conjecture,” Annals Pure Appl. Logic 108, 15–18 (2001).
C. C. Chang and H. J. Keisler, Model Theory (North-Holland Publ. Co., Amsterdam, 1973).
K. Gravett, “Ordered Abelian groups,” Quarterly J.Math., Oxford Second Ser. 7, 57–63 (1956).
H. Hahn, “Über die nichtarchimedischen Grössensystem,” Sitzungsberichte der Kaiserlichen Akademie derWissenschaften,Mathematisch-Naturwissenschaftliche Klasse (Wien) 116, no. Abteilung IIa, 601–655 (1907).
J. van der Hoeven, “Operators on generalized power series,” Illinois J. Math. 45, 1161–1190 (2001).
I. Kaplansky, “Maximal fields with valuations,” Duke Math. J. 9, 303–321 (1942).
S. Kuhlmann and M. Matusinski, “Hardy type derivations on fields of exponential logarithmic series,” J. Algebra 345, 171–189 (2011).
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).
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].
B. H. Neumann, “On ordered division rings,” Transact. Amer.Math. Soc. 66, 202–252 (1949).
A.M. Robert, A Course in p-Adic Analysis, Graduate Texts inMathematics 198 (Springer-Verlag, 2000).
H. Schoutens, The Use of Ultraproducts in Commutative Algebra, Lecture Notes in Mathematics (Springer-Verlag, Berlin, Heidelberg, 2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the author in English.
Rights and permissions
About this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046618040039