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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](http://www.apmaths.uwo.ca/~djeffrey/Offprints/Adamchik.pdf) (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.2011-04-18
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    Yes it would be interesting to have the galois theory of these special functions - or perhaps they can just solve everything?2011-04-18
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    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](http://books.google.com/books?id=jPFaLY31gwYC&pg=SA3-PA261), where he makes use of [Riemann theta functions](http://dlmf.nist.gov/21) to represent roots of algebraic equations. This [MO question](http://mathoverflow.net/questions/23094) might be of interest as well.2011-04-18
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    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...)2011-07-23
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    @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...2011-07-23
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    FWIW @Theo, I retagged those precisely 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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    [Dummit's paper](http://www.ams.org/mcom/1991-57-195/S0025-5718-1991-1079014-X/S0025-5718-1991-1079014-X.pdf) and [Hagedorn's paper](http://dx.doi.org/10.1006/jabr.2000.8428).2011-04-18
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    Is it an open problem??2011-04-18
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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