Solution of a polynomial of degree n with soluble galois group. 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? 

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, ...).
