Computing sums of radicals in polynomial time

For a certain sum of radicals the author presents a Monte Carlo algorithm that runs in polynomial time to decide whether the sum is contained in some number field Q(α), and, if so, its coefficient representation in Q(α) is computed. As a special case the algorithm decides whether the sum is zero. The main algorithm is based on a subalgorithm which is of interest in its own right. This algorithm uses probabilistic methods to check for an element β of an arbitrary (not necessarily) real algebraic number field Q(α) and some positive rational integer r whether there exists an rth root of β in Q(α)