On prolongations of valuations to the composite field

Abstract

Let vbe a Krull valuation of a field Kwith valuation ring Rvand K1, K2be finite separable extensions of Kwhich are linearly disjoint over K. Assume that the integral closure of Rvin the composite field K1K2is a free Rv-module. For a given pair of prolongations v1, v2of vto K1, K2respectively, it is shown that there exists a unique prolongation wof vto K1K2which extends both v1, v2. Moreover with Sias the integral closure of Rvin Ki, if the ring S1S2is integrally closed and the residue field of vis 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 vover the residue field of v. As an application, it is deduced that if K1, K2are algebraic number fields which are linearly disjoint over K=K1∩K2, then the number of prime ideals of the ring AK1K2of algebraic integers of K1K2lying over a given prime ideal ℘of AKequals the product of the numbers of prime ideals of AKilying over ℘for i =1, 2.