Described is a technology, such as implemented in a computational software program, by which a minimal polynomial is efficiently determined for a radical expression over the ring Z of integer numbers or the ring Q of rational numbers. The levels of the radical are grouped into a level permutation group that is used to find a level permutation set. An annihilation polynomial is found based upon the level permutation set. The annihilation polynomial is factored, and a selection mechanism selects the minimal polynomial based upon the annihilation polynomial's factors.