An element \(\psi\) of the field of rational functions \(\mathbb{K}(x_1,\dots, x_d)\) is called closed if the subfield \(\mathbb{K}(\psi)\) is algebraically closed in \(\mathbb{K}(x_1,\dots, x_d)\). For any \(\phi\in \mathbb{K}(x_1,\dots, x_d)\backslash \mathbb{K}\) there is a closed element \(\psi\) and an \(H(t)\in\mathbb{K}(t)\) such that \(H(\psi)=\phi\). We say that \(\psi\) is a generating element for \(\phi\). A subfield \(F<\mathbb{K}(x_1,\dots, x_d)\) is said to be saturated if for every \(\phi\not \in\mathbb{K}\) its generating element lies in \(F\).
In the paper under review, the saturation property is considered for the subfield of invariants \(\mathbb{K}(x_1,\dots, x_d)^G\) of a finite group \(G\) of automorphisms of \(\mathbb{K}(x_1,\dots, x_d)\) in characteristic zero case. The saturation property for the algebra of polynomial invariants \(\mathbb{K}[x_1,\dots, x_d]^G\) has earlier been studied by the authors [Ukr. Math. J. 59, No. 12, 1783–1790 (2007; Zbl 1164.13302)].