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A note on defining transcendentals in function fields

Published online by Cambridge University Press:  12 March 2014

Arno Fehm  and
Wulf-Dieter Geyer
Arno Fehm
Affiliation:
School of Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel, E-mail: [email protected]
Wulf-Dieter Geyer
Affiliation:
Universität Erlangen-Nürnberg, Mathematisches Institut, Bismarckstr. 1 1/2, 91054 Erlangen, Germany, E-mail: [email protected]
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Abstract

The work [11] deals with questions of first-order definability in algebraic function fields. In particular, it exhibits new cases in which the field of constant functions is definable, and it investigates the phenomenon of definable transcendental elements. We fix some of its proofs and make additional observations concerning definable closure in these fields.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009 , pp. 1206 - 1210
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Bélair, Luc and Duret, Jean-Louis, Définissabilité dans les corps de fonctions p-adiques, this Journal, vol. 56 (1991), no. 3, pp. 783–785.Google Scholar
[2]Bergelson, Vitaly and Shapiro, Daniel B., Multiplicative subgroups of finite index in a ring, Proceedings of the American Mathematical Society, vol. 116 (1992), no. 4, pp. 885–896.CrossRefGoogle Scholar
[3]Chatzidakis, Zoé, Simplicity and independence for pseudo-algebraically closed fields, Models and Computability (Cooper, S. Barry and Truss, John K., editors), Cambridge University Press, 1997.Google Scholar
[4]Chevalley, Claude, Introduction to the theory of algebraic functions of one variable, American Mathematical Society, 1951.CrossRefGoogle Scholar
[5]Deuring, Max, Lectures on the theory of algebraic functions of one variable, Lecture Notes in Mathematics 314, Springer, 1973.CrossRefGoogle Scholar
[6]Duret, Jean-Louis, Sur la théorie élémentaire des corps de fonctions, this Journal, vol. 51 (1986), no. 4, pp. 948–956.Google Scholar
[7]Eisenträger, Kirsten and Shlapentokh, Alexandra, Undecidability in function fields ofpositive characteristic, International Mathematics Research Notices, to appear.Google Scholar
[8]Fehm, Arno, Subfields of ample fields I. Rational maps and definability, (2008), arXiv:081l. 2895vl [math. AG].Google Scholar
[9]Harbater, David, On function fields with free absolute Galois groups. Journal für die reine und angewandte Mathematik, to appear.Google Scholar
[10]Junker, Markus and Koenigsmann, Jochen, Schlanke Körper (Slim Fields), this Journal, to appear.Google Scholar
[11]Koenigsmann, Jochen, Defining transcendental in function fields, this Journal, vol. 67 (2002), no. 3, pp. 947–956.Google Scholar
[12]Malcev, A. I., On the undecidability of the elementary theories of certain fields (Russian), Sibirskij Matematiceskij Zurnal, vol. 1 (1960), no. 1, pp. 71–77.Google Scholar
[13]Poonen, Bjorn and Pop, Florian, First-order definitions in function fields over anti-mordellic fields, Model theory with applications to algebra and analysis (Chatzidakis, , Macpherson, , Pillay, , and Wilkie, , editors), Cambridge University Press, 2007.Google Scholar
[14]Pop, Florian, Embedding problems over large fields, Annals of Mathematics, vol. 144 (1996), pp. 1–34.CrossRefGoogle Scholar
[15]Rosenlicht, Maxwell, Automorphisms of function fields, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 1–11.CrossRefGoogle Scholar