While doing an iota worth of digging into deceivingly basic introductions to Galois theory I had this wild thought, since one of the areas it's helpful in is to prove that certain polynomials can't have their roots expressed using a certain set of operations, then how about if we focused our attention on whether we can find all other operations for which that would be possible? Would such set of those operations form a group and behave nicely? More importantly has this been studied before?
(I assume that there exists other operations than "take 5th root" for e.g., otherwise this is a very non-interesting question)