Journal of Symbolic Logic 67 (3):947-956 (2002)

Abstract
Given any field K, there is a function field F/K in one variable containing definable transcendentals 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 $\emptyset-definability$ of K in F is established for any function field F/K in one variable, provided K is large, or $K^{x}\,/(K^{x})^n$ is finite for some integer n > 1 coprime to char K
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DOI 10.2178/jsl/1190150142
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The Undecidability of Pure Transcendental Extensions of Real Fields.Raphael M. Robinson - 1964 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 10 (18):275-282.

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Schlanke Körper (Slim fields).Markus Junker & Jochen Koenigsmann - 2010 - Journal of Symbolic Logic 75 (2):481-500.

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