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Defining transcendentals in function fields

Published online by Cambridge University Press:  12 March 2014

Jochen Koenigsmann
Jochen Koenigsmann*
Affiliation:
Universität Konstanz, Fachbereich Mathematik und Statistik Fach D204, 78457 Konstanz, Germany, E-mail: [email protected]
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Abstract

Given any field K, there is a function field F/K in one variable containing definable transcendental over K, i.e., elements in F / K first-order definable in the language of fields with parameters from K. Hence, the model-theoretic and the field-theoretic relative algebraic closure of K in F do not coincide. E.g., if K is finite, the model-theoretic algebraic closure of K in the rational function field K(t) is K(t).

For the proof, diophantine ∅-definability ofK in F is established for any function field F/K in one variable, provided K is large, or K× /(K×)n is finite for some integer n > 1 coprime to char K.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 3 , September 2002 , pp. 947 - 956
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[1]Bélair, L. and Duret, J.-L., Définissabilité dans les corps de fonctions p-adiques, this Journal, vol. 56 (1991), no. 3, pp. 783–785.Google Scholar
[2]Bélair, L. and Duret, J.-L., Indécidabilité des corps de courbe réelle, this Journal, vol. 59 (1994), no. 1, pp. 87–91.Google Scholar
[3]Bergelson, V. and Shapiro, D. 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
[4]Chatzidakis, Z., Dries, L. V. D., and Macintyre, A., Definable sets over finite fields, Journal für die Reine und Angewandte Mathematik, vol. 427 (1992), pp. 107–135.Google Scholar
[5]Chevalley, C., Introduction to the theory of algebraic functions of one variable, AMS Mathematical Surveys VI, 1951.CrossRefGoogle Scholar
[6]Delon, F., Separably closed fields, Model Theory and Algebraic Geometry (Bouscaren, E., editor), Lecture Notes in Mathematics, vol. 1696, Springer, 1998, pp. 143–176.CrossRefGoogle Scholar
[7]Duret, J.-L., Sur la théorie élémentaire des corps de fonctions, this Journal, vol. 51 (1986), no. 4, pp. 948–965.Google Scholar
[8]Fulton, W., Algebraic Curves, Benjamin, 1969.Google Scholar
[9]Hurwitz, A., Analytische Gebilde mit eindeutigen Transformationen in sich, Mathematische Annalen, vol. 41 (1893), pp. 403–422.CrossRefGoogle Scholar
[10]Iwasawa, K. and Tamagawa, T., On the group of automorphisms of a function field, Journal of the Mathematical Society of Japan, vol. 3 (1951), no. 1, pp. 137–147, and also in vol.4 (1952), pp. 100–101.CrossRefGoogle Scholar
[11]Matiyasevich, Y., Enumerable sets are Diophantine, Soviet Mathematics Doklady, vol. 11 (1970), pp. 354–357.Google Scholar
[12]Poincaré, H., Sur un théorème de M. Fuchs, Acta Mathematica, vol. 7 (1885), pp. 1–10.CrossRefGoogle Scholar
[13]Pop, F., Embedding problems over large fields, Annals of Mathematics, vol. 144 (1996), pp. 1–34.CrossRefGoogle Scholar
[14]Robinson, J., Definability and decision problems in arithmetic, this Journal, vol. 14 (1949), pp. 98–114.Google Scholar
[15]Robinson, R., The undecidability of pure transcendental extensions of real fields, Zeitschrift für Mathenatische Logik und Grundlagen der Mathematik, vol. 10 (1964), pp. 275–282.CrossRefGoogle Scholar
[16]Roquette, P., Über die Automorphismengruppe eines algebraischen Funktionenkörpers, Archiv der Mathematik, vol. III (1952), pp. 343–350.CrossRefGoogle Scholar
[17]Rosenlicht, M., Automorphisms offunction fields, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 1–11.CrossRefGoogle Scholar
[18]Rumely, R., Undecidability and definability for the theory of global fields, Transactions of the American Mathematical Society, vol. 262 (1980), no. 1, pp. 195–217.CrossRefGoogle Scholar
[19]Scharlau, W., Quadratic and hermitian forms, Springer, 1985, Grundlehren270.CrossRefGoogle Scholar
[20]Schmid, H. L., Über die Automorphismen eines algebraischen Funktionenkörpers von Primzahlcharakteristik, Journal für die Reine und Angewandte Mathematik, vol. 179 (1938), pp. 5–15.CrossRefGoogle Scholar
[21]Silverman, J. H., The arithmetic of elliptic curves, Springer, Heidelberg, 1986.CrossRefGoogle Scholar
[22]Stichtenoth, H., Algebraic function fields and codes, Springer, Heidelberg, 1993.Google Scholar
[23]Weierstraß, K., Aus einem noch nicht veröffentlichten Briefe an Herrn P. Schwarz, Werke Bd. II, pp. 235–244.Google Scholar