TITLE: How to solve polynomials?
QUESTION [12 upvotes]: 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

REPLY [8 votes]: Solving polynomials of higher degree that are solvable by radicals is a hard problem and there are no general formula. There are many approaches, but most rely on the concept of a Galois resolvent which is an auxiliary polynomial that factors if the original polynomial is solvable. The following papers might be useful:
Solving Solvable Quintics
D. S. Dummit
Mathematics of Computation
Vol. 57, No. 195 (Jul., 1991), pp. 387-401 
General Formulas for Solving Solvable Sextic Equations*1
Thomas R. Hagedorn
Journal of Algebra
Volume 233, Issue 2, 15 November 2000, Pages 704-757
On solvable septics
LAU JING FENG
http://scholarbank.nus.edu.sg/handle/10635/14460