This paper presents a new, simple, and efficient algorithm for factoring polynomial's in several variables over an algebraic number field. The algorithm is then used iteratively to construct the splitting field of a polynomial over the Integers. Finally the factorization and splitting field algorithms are applied to the problem of determining the transcendental part of the integral of a rational function. In particular, a constructive procedure is given for finding the least degree extension field in which the Integral can be expressed.