How to denest Ramanujan's nested radicals

The author presents a simple condition when nested radical expressions of depth two can be denested using real radicals or radicals of some bounded degree. He describes the structure of these denestings and determines an upper bound on the maximum size of a denesting. Also for depth two radicals he describes an algorithm that will find such a denesting whenever one exists. Unlike all previous denesting algorithms the algorithm does not use Galois theory. In particular, he avoids the construction of the minimal polynomial and splitting field of a nested radical expression. Thus he can obtain the first denesting algorithm whose run time is at most, and in general much less, than polynomial in description size of the minimal polynomial. The algorithm can be used to determine non-trivial denestings for expressions of depth larger than two.