On prolongations of valuations to the composite field

Abstract

Let v be a Krull valuation of a field K with valuation ring Rv and K1,K2 be finite separable extensions of K which are linearly disjoint over K. Assume that the integral closure of Rv in the composite field K1K2 is a free Rv-module. For a given pair of prolongations v1,v2 of v to K1,K2 respectively, it is shown that there exists a unique prolongation w of v to K1K2 which extends both v1,v2. Moreover with Si as the integral closure of Rv in Ki, if the ring S1S2 is integrally closed and the residue field of v is perfect, then f(w/v)=f(v1/v)f(v2/v), where f(v′/v) stands for the degree of the residue field of a prolongation v′ of v over the residue field of v. As an application, it is deduced that if K1,K2 are algebraic number fields which are linearly disjoint over K=K1∩K2, then the number of prime ideals of the ring AK1K2 of algebraic integers of K1K2 lying over a given prime ideal ℘ of AK equals the product of the numbers of prime ideals of AKi lying over ℘ for i=1,2.