On the Saturation of Subfields of Invariants of Finite Groups

Every subfield $\mathbb K(\phi)$ of the field of rational fractions $\mathbb K(x_1,\dots,x_n)$ is contained in a unique maximal subfield of the form $\mathbb K(\psi)$. The element $\psi$ is said to be generating for the element $\phi$. A subfield of $\mathbb K(x_1,\dots,x_n)$ is said to be saturated if, together with every its element, the subfield also contains the generating element. In the paper, the saturation property is studied for the subfields of invariants $\mathbb K(x_1,\dots,x_n)^G$ of a finite group $G$ of automorphisms of the field $\mathbb K(x_1\dots,x_n)$.