An affine version of a theorem of Nagata

Let R be an affine k-domain over the field k. The paper's main result is that, if R admits a non-trivial embedding in a polynomial ring K[s] for some field K containing k, then R can be embedded in a polynomial ring F[t] which extends R algebraically. This theorem can be applied to subrings of a ring which admits a non-zero locally nilpotent derivation. In this way, we obtain a concise new proof of the cancellation theorem for rings of transcendence degree one for fields of characteristic zero.