Automorphism groups of ruled function fields and a problem of Zariski

Abstract:

Let ${K_1}$ and ${K_2}$ be finitely generated extensions of a field $K$ and let $x$ be transcendental over ${K_1}$ and ${K_2}$, and assume ${K_1}(x) = {K_2}(x)$. The main results show that if $K$ is infinite and the group of automorphisms of ${K_2}$ over $K$ is finite, or if $K$ is finite and the group of automorphisms of $\bar K{K_2}$ over $\bar K$ ($\bar K$ the algebraic closure of $K$) is finite, then ${K_1}$ equals ${K_2}$.