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Background: Given the fundamental theorem of algebra every polynomial of degree n has n roots. From Galois Theory we know that we can only find exact solutions of polynomials if their corresponding Galois group is soluble. I am studying Galois Theory ( Ian Stewart ) and I am not getting the result out of it that I expected. I expected to learn to determine for a polynomial of degree n its corresponding Galois group, and if it that group is soluble a recipe to find the exact roots of that polynomial. My experience thus far with Galois Theory is that it proves that there is no general solution for a polynomial of degree 5 and higher.

Question: I want to learn to solve polynomials of degree 5 and higher if they have a corresponding soluble Galois group. From which book or article can I learn this?

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By exact roots you probably mean radical expressions. Even for equations whose Galois group is unsolvable there might be exact trigonometric expressions for the roots.

If you know German, the diploma thesis "Ein Algorithmus zum Lösen einer Polynomgleichung durch Radikale" (An algorithm for the solution of a polynomial equation by radicals) by Andreas Distler is exactly what you're looking for. It is available online. It also contains several program codes.

On the other hand, today there are many computer algebra systems which can compute the Galois group of a given polynomial or number field (GAP, Sage, ...).

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  • $\begingroup$ This article is from 2005, is the knowledge I am looking for as recent as 2005?!! Why do the books about Galois Theory say absolute nothing about this subject? ( I have seen, browsed through many Galois Theory books, they all use the same X^3 + 2 = 0 example, and so on. ) $\endgroup$ Jan 9 '12 at 12:26
  • $\begingroup$ I was thinking, if English is not the language of mathematics, then what treasures are hidden from us in the Chinese ( Spanish, French, German, ... ) literature? $\endgroup$ Jan 9 '12 at 12:39
  • $\begingroup$ Basically Andreas Distler has observed that all the proofs in Galois theory can be made constructive; surely this was already known for many decades. But often this leads to a large system of linear equations which is not really nice to do by hand. See for example his algorithm for the primitive element. I am sure that there are many books about "computational Galois theory". $\endgroup$ Jan 10 '12 at 10:42

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