Using Galois theory we can effectively compute whether or not a polynomial is solvable in radicals - technically this means you can build a chain of field extensions by adding $n$-th roots of previously defined elements.
Anyway I was wondering, how do we actually solve the polynomials when they can be solved?
I have some ad-hoc methods to solve quadratic, general cubic and quartic as well as Gauss method to express some primitive roots of unity but I would like to read about something more general.
Also I would be interested in any other objects than radicals that are studied like exponential sums can be used to solve a smaller set of polynomials for example.
Related Galois groups of polynomials and explicit equations for the roots