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