Starting with the field of rational numbers, we can consider the extension field obtained by adjoining all possible radicals (i.e., roots of polynomials Xⁿ – c). By repeating this, we obtain a hierarchy of radical extensions, that poses many interesting algorithmic questions. I will consider some of those in this talk, including special cases of the denesting problem, that asks for the least field in the hierarchy a given nested radical will reside in. One such case was considered by Ramajuan and relates to generators for abelian fields.

When dealing with radicals in Magma, one has to deal with problems of ambiguity due to multivaluedness of radicals. I will propose some solutions.