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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: Jochen.Koenigsmann@uni-konstanz.de
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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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