This dissertation presents methods for determining the behavior of prime ideals m an integral extension of a Dedekind domain. One tool used to determine this behavior is an algorithm that computes which prime ideals ramify in a finite separable extension. Other results about factorization of prime ideals are improved and applied to finite extensions. By considering a set of finite extensions whose union is an infinite extension, it is possible to predict ideal factorization in the infinite extension as well. Among other things, this ideal factorization determines whether a given infinite extension is almost Dedekind. These methods and results yield some interesting facts when they are demonstrated on a pair of classical rings of algebraic number theory.