On closed rational functions in several variables

Abstract

Let K = K¯ be a field of characteristic zero. An element ϕ ∈ K(x1,... ,xn) is called a closed rational function if the subfield K(ϕ) is algebraically closed in the field K(x1,... ,xn). We prove that a rational function ϕ = 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 ϕ = f/g is closed if and only if the pencil αf + βg contains only finitely many reducible hypersurfaces. Some sufficient conditions for a polynomial to be irreducible are given.