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Schlanke Körper (Slim fields)

Published online by Cambridge University Press:  12 March 2014

Markus Junker  and
Jochen Koenigsmann
Markus Junker
Affiliation:
Mathematisches Institut, Eckerstrasse 1, 79104 Freiburg, Germany. E-mail: markus.junker@math.uni-freiburg.de
Jochen Koenigsmann
Affiliation:
Mathematical Institute, 24-29 St Giles', Oxford Ox1 3Lb, UK. E-mail: koenigsmann@maths.ox.ac.uk
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Abstract

We examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 2 , June 2010 , pp. 481 - 500
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1]Adler, Hans, Explanation of independence, Ph.D. thesis, Albert–Ludwigs–Universität Freiburg, 2005, http://arxiv.org/abs/math.L0/0511616.Google Scholar
[2]Ax, James, The elementary theory of finite fields, Annals of Mathematics, vol. 88 (1968), pp. 239–271.CrossRefGoogle Scholar
[3]Chatzidakis, Zoe, van den Dries, Lou, and Macintyre, Angus, Definable sets over finite fields, Journal für die Reine und Angewandte Mathematik, vol. 427 (1992), pp. 107–135.Google Scholar
[4]van den Dries, Lou, Elimination theory for the ring of algebraic integers, Journal für die Reine und Angewandte Mathematik, vol. 388 (1988), pp. 189–205.Google Scholar
[5]Engler, Antonio Jose and Prestel, Alexander, Valued Fields, Springer, 2005.Google Scholar
[6]Fehm, Arno, Subfields of ample fields I. Rational maps and definability, 2008, preprint available at arXiv:0811.2895.Google Scholar
[7]Harnik, Victor and Harrington, Leo. Fundamentals of forking, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 245–286.CrossRefGoogle Scholar
[8]Jarden, Moshe and Wheeler, William H., Model-complete theories of e-free Ax fields, this Journal, vol. 48 (1983), no. 4, pp. 1125–1129.Google Scholar
[9]Junker, Markus, A note on equational theories, this Journal, vol. 65 (2000), no. 4, pp. 1705–1712.Google Scholar
[10]Junker, Markus and Kraus, Ingo, Theories with equational forking, this Journal, vol. 67 (2002), no. 1, pp. 326–340.Google Scholar
[11]Kim, Byunghan and Pillay, Anand, Simple theories, Annals of Pure and Applied Logic, vol. 88 (1997), no. 2-3, pp. 149–164.CrossRefGoogle Scholar
[12]Koenigsmann, Jochen, Defining transcendentals in function fields, this Journal, vol. 67 (2002). no. 3, pp. 947–956.Google Scholar
[13]Onshuus, Alf, Properties and consequences of thorn-independence, this Journal, vol. 71 (2006), no. 1, pp. 1–21.Google Scholar
[14]Pas, Johan, Uniform p-adic cell decomposition and local zeta functions, Journal für die Reine und Angewandte Mathematik, vol. 399 (1989), pp. 137–172.Google Scholar
[15]Pop, Florian. Embedding problems over large fields, Annals of Mathematics, vol. 144 (1996). no. 1, pp. 1–34.CrossRefGoogle Scholar
[16]Prestel, Alexander and Ziegler, Martin, Model theoretic methods in the theory of topological fields, Journal für die Reine und Angewandte Mathematik, vol. 299/300 (1978), pp. 318–431.Google Scholar
[17]Robinson, Julia, The undecidability of algebraic rings and fields, Proceedings of the American Mathematical Society, vol. 10 (1959), pp. 950–957.CrossRefGoogle Scholar
[18]Robinson, Julia, The decision problem for fields, Proceedings of the International Symposium in Theory of Models (Berkeley), North-Holland (Amsterdam), 1963, pp. 299–311.Google Scholar
[19]Wheeler, William H., Model complete theories of pscudo-algebraically closed fields, Annals of Mathematical Logic, vol. 17 (1979), no. 3, pp. 205–226.CrossRefGoogle Scholar