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. (shrink)

We examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example, they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.