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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

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    FWIW @Theo, I retagged those pre$c$isely because they were contaminating my search results... :)2011-07-23

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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

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    It's an open problem in the sense that it has not been completely solved in cases n>7. But as the above articles show, people generally have a good idea of how it can be done - but it is just so computationally hard.2011-04-18