# Galois groups of polynomials and explicit equations for the roots

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?

• 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, 2011 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.
– user961
Mar 20, 2011 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, 2011 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, 2011 at 5:03
• @KCd: You are welcome to post your comment as an answer so I can close the question.
– user961
Mar 20, 2011 at 5:08