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Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the polynomial in terms of its coefficients assuming the galois group of the polynomial is solvable?

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Some examples of this might be worked out in the book "Galois Theory" by D. Cox. But why do you really want to do this? The reality is that Galois theory is not really designed to make that task straightforward or efficient. In fact Galois made that point himself in one of his letters. If I can dig up that citation I'll post it here later. – KCd Mar 20 '11 at 3:29
@KCd: It seems odd to motivate the subject by saying things about roots of equations and how they can be expressed in terms of the coefficients if and only if the group is solvable and not have a single worked out example. During my studies I haven't found an example so that's why I asked. – davidk01 Mar 20 '11 at 4:40
Ah, you have never seen even one example. In Dummit & Foote's Abstract Algebra they have an exercise which works out explicit formulas for the 17th roots of unity in terms of nested radicals. And you can find Galois-theoretic derivations of the cubic and quartic formulas by a simple Google search for "galois theory cubic formula". I personally think it's far more interesting that you can use Galois extensions to create 9-dimensional division rings (over the rational numbers) since the only division ring most non-algebraists have heard of, namely the real quaternions, has dimension 4. – KCd Mar 20 '11 at 5:01
And since you write about not seeing an example in your studies, why not approach a local professor (perhaps the teacher of your algebra course) and ask in person how to get such examples? – KCd Mar 20 '11 at 5:03
@KCd: You are welcome to post your comment as an answer so I can close the question. – davidk01 Mar 20 '11 at 5:08
up vote 7 down vote accepted

One of the fundamental techniques for doing this is to compute the (Lagrange) resolvants associated to the equation. Assuming the group is solvable, the resolvants will be factorizable and amenable to lower degree auxiliary equations. To have a first idea of all this, I would advice you to look at Harold Edwards splendid book : "Galois Theory" (ISBN 038790980X) where it does it for the elementary cases and an example of the cyclotomic equation.

With modern Computer Algebra System, one can more easily explore the idea further, but considerable ingenuity is needed because the degree of the auxiliary equations rise quickly. Some CAS (such as GAP which is a free and open-source academic research tool) are able to give you the Galois groups of low degree polynomials, as well as properties of splitting fields and you can look at their tutorial to have examples of their use.

Effective and Inverse Galois Theory is still an active research subject. Some of the works by Annick Valibouze, N. Durov, Klueners, etc. might help you to go further.

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The book Classical Galois theory: with examples by Gaal contains explicit computations, if not a complete algorithm.

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I feel the book "Algebraic theories" by Dickson(now its coming in Dover phoenix edition) have some "Real" stuff about Galois theory in the sense that answers your question of explicit calculations.

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