On generalized cancellation problem

Abstract

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] (K1 (t)) implies an isomorphism of K1 and K1. In this paper, we address a related problem: whether the ring (field) embedding of K1 [t] (K1 (t)) into K2 [t] (K1 (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 K1. As a consequence, we answer a question of Abhyankar, Eakin and Heinzer [J. Algebra 23 (1972) 310-342]. © 2004 Elsevier Inc. All rights reserved.