Embeddings of function fields into ample fields (Q625509)

Let \(K\) be a field, \(V\) be a smooth \(K\)-variety and \(E\) be an ample field containing \(K.\) The author gives a necessary and sufficient condition for the existence of \(K\)-embedding of the function field \(K(V)\) into \(E\). The number of such embeddings is also computed. The author also investigates some aspects of the subfield structure of ample fields which leads to new examples of non-ample fields. Some connections between the existence of rational places on function fields and embeddings into ample fields are obtained; this connects the paper to an interesting problem raised by \textit{F. V. Kuhlmann} in [Adv. Math. 188, No. 2, 399--424 (2004; Zbl 1134.12304)], several equivalent versions of which are given.