# Using Group Theory

It is my first post here.

I'm studying Group Theory and I found lots of examples of it but for advanced applications.

What I'm trying to find or understand is just the opposit! I want to use Group Theory to solve quadratic polynomials, for example. Can someone please provide some link or example of how to apply Group Theory to solve simple things like to find the roots of a polynomial of degree $n \leq 4$ ? A simple: $x^2+x+c=0$ would be enough to let me start to use it in GAP and Sage Package.

Thank you in advance.

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Why exactly do you expect group theory to be of help in solving quadratic equations? Now... if someone found how to have it make coffee, that would be useful! – Mariano Suárez-Alvarez May 11 '12 at 21:04
Here's the thing: polynomials are essentially a "ring concept", you need to add AND multiply. Groups are a "one operation structure" better suited to investigating situations where we have just one operation (like, for example, invertible functions under composition). For an application of groups to "simple things" you might want to look into something like Burnside's Lemma. – David Wheeler May 11 '12 at 22:41
You say "Why not the so powerful Group Theory for a simple task?"... What would you think if someone asked you how to use a bulldozer to sort alphabetically a list of names of people? Bulldozers are pretty powerful tools too! – Mariano Suárez-Alvarez May 11 '12 at 22:47
@Mariano Popular expositions frequently mention Galois theory's use of group theory and symmetry for solving polynomial equations, but without giving any details. This naturally raises questions (such as above) as to how those methods work in simple cases. But, of course, such expositions do not motivate questions about group-theoretical coffee-making or sorting bulldozers. Please keep in mind that it might take great courage for students to pose their first questions here. Let's strive to handle these questions mathematically, rather than jokingly. – Bill Dubuque May 12 '12 at 23:28
Dear Bill, if you read the question of my first comment you will notice that (apart from making a silly joke) I was asking the OP where he got the idea. At the time, I did not think it useful, and I do not think it useful now, to speculate on where the confusion arose: I asked in order to, informedly, explain it away. – Mariano Suárez-Alvarez May 12 '12 at 23:40

There are several issues with your question.

First, the meaning of "simple things" will be different for almost any two persons.

Second, you seem to have decided on your own that Group Theory has to be useful to find roots of polynomials, which it isn't, at least in a naive sense. For that matter, you could be asking to use Measure Theory to find roots of polynomials, and the question would be equally meaningless.

Third, you are definitely not grasping what Group Theory is about. The big merit of Group Theory is precisely that it allows one to consider many apparently unrelated operations (numerical operations, geometric transformations, permutations, to name the most common) under the same light in a unified framework.

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I'll leave this site. Trying to search how to close this account. Thank you for those who sincery tried to help me but I don't like to talk with some kids here. – Luiz Roberto Meier May 11 '12 at 23:20
And, by the way: several of the "kids" here are professional mathematicians. It's hard to imagine who you expected to talk with. – Martin Argerami May 11 '12 at 23:49
@LuizRobertoMeier: I am a professional mathematician; and I have even taught courses on Galois Theory, though it is not really my area of expertise. If you can get so much knowledge from Google, please teach us how to solve equations using Galois Theory. We'll be waiting. – Martin Argerami May 12 '12 at 1:59
I just bought a book about Galois Theory and another about Abstract Algebra from Amazon. I'll prove it to you that it is possible to solve polynomials with any degree, also, I'll use Carlyle diagrams and tables to make it easy to you to follow. Professionals don't behave in this way with someone who just join a site. Your name and details is saved. Need time to read both books but I will do it. I like to unmask people like you. No more time to lose in this thread for now. – Luiz Roberto Meier May 12 '12 at 2:07
@LuizRobertoMeier Yes, "Galois theory can be used to solve polynomials of any degree", but you disregard the fact that "solve" does not mean explicit solution and "used" involves much more simple algebraic manipulation than the Galois-free usual solution formula. Asking people for help in "solving quadratic equations by Galois theory" and then threatening them because you cannot envision that your understanding of group theory is less than perfect does not speak well of you. You don't even entertain the possiblity that several professional mathematicians could have a point. – Phira May 13 '12 at 7:36