Abstract. Radical simplification is an important part of symbolic computation systems. Until now no algorithms were known for the general denesting problem. If the base field contains all roots of unity, then necessary and sufficient conditions for a denesting are given, and the algorithm computes a denesting of alpha when it exists. If the base field does not contain all roots of unity, then it is shown how to compute a denesting that is within one of optimal over the base field adjoining a single root of unity. Throughout this paper, a primitive l-th root of unity is respresented by its symbol j_l rather than as a nested radical. The algorithms require computing the splitting field of the minimal polynomial of alpha over k, and have exponential running time.