A note on defining transcendentals in function fields

The work [11] deals with questions of first-order definability in algebraic function fields. In particular, it exhibits new cases in which the field of constant functions is definable, and it investigates the phenomenon of definable transcendental elements. We fix some of its proofs and make additional observations concerning definable closure in these fields. Introduction. The model theoretic notion of algebraic closure is in some sense a generalization of the classical algebraic notion of relative algebraic closure in fields. Although for fields in some cases these two notions coincide, their relation in general is more complicated: The main result of Koenigsmann's work [1 1] is that every field K has an extension F such that the model theoretic algebraic closure of K in F does not coincide with its relative algebraic closure in F . This result is achieved by studying definable sets in algebraic function fields, and an important ingredient in the proof is the discovery of a huge class of function fields in which the field of constant functions is definable. Unfortunately, some of the proofs of [1 1] are erroneous. The main purpose of this note is to fix these proofs. The first section deals with the definability of the field of constants: The formula given in [11] for defining the field of constants in case the index of its w-th powers is finite does not fulfill its purpose. We give a different formula to prove definability of the field of constants in that case. The second section handles the main result mentioned above: By giving a counterexample we show that a crucial step in the proof works well in the case of a perfect field of constants but fails in general. We fix that gap. In the third section we observe that besides the cases Koenigsmann mentions, each inseparable function field has the property that model theoretic algebraic closure and relative algebraic closure do not coincide. Notation. We recall some of the notions and definitions from [11]: A function field of one variable is a finitely generated extension of fields F \ K of transcendence degree one such that the field of constants K is relatively algebraically closed in F . If furthermore the extension F\K is separable and thus regular, we call it a separable function field of one variable. The genus of F\K is denoted by Received April 21, 2008. The authors would like to thank Lior Bary-Soroker and Moshe Jarden for helpful comments on a previous version. This work was supported by the European Commission under contract MRTN-CT2006-035495. © 2009. Association for Symbolic Logic 0022-48 1 2/09/7404-0007/$ 1 .50