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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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With quintics, for instance, you need either theta/elliptic functions or hypergeometric functions in the general case to analytically represent the roots, but I find the symbolic expressions too unwieldy. In general, one tack is to find substitutions akin to the Tschirnhausen substitution (which in a sense is a generalization of the "depression" substitution $x=u-\frac{b}{na}$ for the polynomial $ax^n+bx^{n-1}+\dots$) to bring the polynomial to a more manageable form. –  Ｊ. Ｍ. Apr 18 '11 at 10:36
Yes it would be interesting to have the galois theory of these special functions - or perhaps they can just solve everything? –  quanta Apr 18 '11 at 10:38
As you might know, the higher you go in degree, the more special functions you need to add to your repertoire. See for instance Umemura's paper here, where he makes use of Riemann theta functions to represent roots of algebraic equations. This MO question might be of interest as well. –  Ｊ. Ｍ. Apr 18 '11 at 10:55
Sorry @Theo, that was only five or so questions so I didn't think much of it. I think that's that. (If it were more than that I'd have restrained myself suitably...) –  Ｊ. Ｍ. Jul 23 '11 at 18:06
@J.M. Sorry, I'm having a bad day today and shouldn't have complained... It's just that Willie already did two or three re-tags today and I'm having trouble finding stuff... –  t.b. Jul 23 '11 at 18:12
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up vote 6 down vote accepted

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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Dummit's paper and Hagedorn's paper. –  Ｊ. Ｍ. Apr 18 '11 at 12:09
Is it an open problem?? –  quanta Apr 18 '11 at 16:10
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. –  Bonanza Apr 18 '11 at 20:31
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