On generalized cancellation problem

A well-known cancellation problem of Zariski asks whether for two given domains (fields) K1 and K2, an isomorphism of K1[t] (K(t)) and K2[t] (K2(t)) implies an isomorphism of K1 and K2. In this paper, we address a related problem: whether the ring (field) embedding of K1[t] (K1(t)) into K2[t] (K2(t)) implies the ring (field) embedding of K1 into K2? Our main result is affirmative: if K1 and K2 are arbitrary domains (fields) of the finite transcendence degree and K1[t] (K1(t)) can be embedded into K2[t] (K2(t)) then K1 can be embedded into K2. As a consequence, we answer a question of Abhyankar, Eakin and Heinzer [J. Algebra 23 (1972) 310–342].