The author describes an algorithm for factoring polynomials over arbitrary number fields. This algorithm works as follows. Given a polynomial f, defined over a number field K. We take the norm N f of f to \({\mathbb{Q}}[X]\), and factor N f over \({\mathbb{Q}}\). If N f is square free we derive a factorization of f. The author gives an explicit value for the time complexity of this algorithm if in the factorization over \({\mathbb{Q}}\) the Lenstra-Lenstra-Lovász algorithm is used. This complexity is polynomial in the degree of f, the degree of K and the logarithm of the discriminant of K. The condition that N f must be square free is not a strong one, the author shows that it is easy to transform a given polynomial in such a way that the norm becomes square free.