Defining transcendentals in function fields

Abstract

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 of K 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 .