The ruled residue theorem for simple transcendental extensions of valued fields

Abstract

A proof is given for the Ruled Residue Conjecture, which asserts that if υ \upsilon is a valuation of a simple transcendental field extension K 0 ( x ) {K_0}(x) and υ 0 {\upsilon _0} is the restriction of υ \upsilon to K 0 {K_0} , then the residue field of υ \upsilon is either ruled or algebraic over the residue field of υ 0 {\upsilon _0} .