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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?

Kenny
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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. –  Mar 07 '12 at 12:44
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    @KannappanSampath, there are solvable polynomial equations of all degrees... – Mariano Suárez-Álvarez Mar 07 '12 at 12:45

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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.

Mariano Suárez-Álvarez
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  • I wrote quintic but this of course applies to all degrees. – Mariano Suárez-Álvarez Mar 07 '12 at 12:56
  • 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 07 '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-Álvarez Mar 07 '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 07 '12 at 20:57
  • My last comment is directed at Mariano. – Kenny Mar 07 '12 at 21:02
  • 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!) – Mariano Suárez-Álvarez Mar 07 '12 at 21:11