On generalized cancellation problem

Abstract

A well-known cancellation problem of Zariski asks whether for two given domains (fields) K 1 and K 2 , an isomorphism of K 1 [ t ] ( K ( t ) ) and K 2 [ t ] ( K 2 ( t ) ) implies an isomorphism of K 1 and K 2 . In this paper, we address a related problem: whether the ring (field) embedding of K 1 [ t ] ( K 1 ( t ) ) into K 2 [ t ] ( K 2 ( t ) ) implies the ring (field) embedding of K 1 into K 2 ? Our main result is affirmative: if K 1 and K 2 are arbitrary domains (fields) of the finite transcendence degree and K 1 [ t ] ( K 1 ( t ) ) can be embedded into K 2 [ t ] ( K 2 ( t ) ) then K 1 can be embedded into K 2 . As a consequence, we answer a question of Abhyankar, Eakin and Heinzer [J. Algebra 23 (1972) 310–342].