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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](http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem). 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](http://www.isibang.ac.in/~sury/galoisreso.pdf) exposition pretty interesting.2012-03-07
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    @KannappanSampath, there are solvable polynomial equations of all degrees...2012-03-07

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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.2012-03-07
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    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?2012-03-07
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    @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.2012-03-07
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    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.2012-03-07
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    My last comment is directed at Mariano.2012-03-07
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    If a rational polynomial has all its roots rational, its Galois group is trivial and you are not going to get anything out of it (in fact, an extremely, utmostly silly way of checking that a rational polynomial has all its roots rational is to run an algorithm to compute its Galois group and seeing if the result is trivial or not!)2012-03-07