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Does galois theory actually have some involvement in solving a solvable quintic, or does it just tell you whether it IS solvable or not?

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Impossibility of solving a general quintic or higher degree equations using radicals is proved in Abel-Ruffini. So, Galois Theory cannot disprove that and hence also disprove Godel's inconsistency Theorem. So, all it can tell you is if a given quintic is solvable by radicals or not and further deduce the roots. You will find this exposition pretty interesting. –  user21436 Mar 7 '12 at 12:44
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@KannappanSampath, there are solvable polynomial equations of all degrees... –  Mariano Suárez-Alvarez Mar 7 '12 at 12:45
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up vote 3 down vote accepted

When the quintic is solvable, one can use the structure of the Galois group to explicitely construct the solutions. It is an immensely impractical task, though!

GAP has a package called RadiRoot which does precisely this.

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I wrote quintic but this of course applies to all degrees. –  Mariano Suárez-Alvarez Mar 7 '12 at 12:56
    
Thanks for the link. –  Kenny Mar 7 '12 at 13:25
    
I think I was told that you have to know the roots (solutions) in order to know the Galois group of a polynomial. That's bad information isn't it? –  Kenny Mar 7 '12 at 18:39
    
@Kenny, indeed, that is not true. There are a few ways to determine the Galois group without knowing the roots. Google for Lagrange resolvents, for example. –  Mariano Suárez-Alvarez Mar 7 '12 at 19:07
    
You say that you can use Galois theory to construct the solutions of any solvable polynomial. Agree? Polynomials with rational ROOTS are always solvable. Agree? So can Galois theory be used for the construction of rational roots? I want to know YOUR opinion because you seem to know what you are talking about and other people are giving me seemingly contradictory information. –  Kenny Mar 7 '12 at 20:57
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