Fields defined by locally nilpotent derivations and monomials

We prove a necessary and sufficient condition for certain fields defined by locally nilpotent derivations and monomials to be algebraically closed in a rational function field. This implies that a counterexample to the Fourteenth Problem of Hilbert in dimension four, which was recently given by the author, is obtained as the kernel of a derivation. It was previously unknown only in dimension four whether there exists such a counterexample.