On closed rational functions in several variables

Let $\mathbb{K}=\bar{\mathbb K}$ be a field of characteristic zero. An element $\varphi\in\mathbb K(x_1,\dots,x_n)$ is called a closed rational function if the subfield $\mathbb K(\varphi)$ is algebraically closed in the field $\mathbb K(x_1,\dots,x_n)$. We prove that a rational function $\varphi=f/g$ is closed if $f$ and $g$ are algebraically independent and at least one of them is irreducible. We also show that a rational function $\varphi=f/g$ is closed if and only if the pencil $\alpha f+\beta g$ contains only finitely many reducible hypersurfaces. Some sufficient conditions for a polynomial to be irreducible are given.