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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.

Arturo Magidin
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Luiz Meier
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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-Álvarez May 11 '12 at 21:04
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    Please list some of the "advanced applications" you encountered so that we don't bother to suggest them to you. I can't really think of how group theory could be applied to quadratic equations. Galois theory does relate solutions of equations to each other using groups, but this would most likely be described as "advanced". – rschwieb May 11 '12 at 21:14
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    Why not to have an example of a quadratic solution using Group Theory just because we ca use simple algebra for that? So, because I can have exactly solutions in Reals or Imaginary I can't use, for example, Newton approximation methods to demonstrate the use of this method? Why not the so powerful Group Theory for a simple task? – Luiz Meier May 11 '12 at 21:37
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    rschwieb: don't put words in my mouth. You do not give an example of Galois showing a step-by-step solution for a simple quadratic equations. If you want some list of 'advanced' usage of Group Theory, type in google: group theory and quantum mechanics or state Y(3,1) and SU(3), gauge and color theory. If you think it is easy to understand, then, you can provide a simple example of Group Theory usage. – Luiz Meier May 11 '12 at 22:00
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    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
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    Got it , David Wheeler. – Luiz Meier May 11 '12 at 22:46
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    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-Álvarez May 11 '12 at 22:47
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    -1: Usually I wouldn't downvote a first-time post, but I was unhappy with the OP's rudeness towards Martin, who bothered to try to help. I have no problem at all with discouraging such people from continuing to use the site. – Tara B May 12 '12 at 10:11
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    @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
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    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-Álvarez May 12 '12 at 23:40
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    @LuizRobertoMeier I do not understand your negative reaction at all. You wrote "only advanced applications": there were no words put in anyone's mouth. My request to know what you were thinking of was meant in all seriousness, not as sarcasm as you treated it. In the future please assume the good-faith of other posters. – rschwieb May 18 '12 at 20:48
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    @LuizRobertoMeier: Something that may be interesting to you is Lagrange's work on the Theory of Equations, in which he used "symmetries" to obtain a more or less uniform understanding of the solutions of the quadratic, cubic, and quartic. Not Galois Theory exactly, but a precursor to Galois Theory. – André Nicolas Oct 07 '13 at 17:57

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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.

Martin Argerami
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  • I have not decided anything. I'm just new here and new to the Group Theory. So, I must understand/study Galois Theory for solution of algebraic equations? – Luiz Meier May 11 '12 at 22:48
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    There's never any need and/or use of Galois theory to solve algebraic equations: all solutions known (degrees 1 to 4) are just arithmetic. The big contribution of Galois Theory is the proof that there is not general solution for degree 5 and above. – Martin Argerami May 11 '12 at 23:49
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    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
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    @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 01:59
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    Polynomial rings. – Luiz Meier May 12 '12 at 02:10
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    By the way, had you searched, you would have found these two answers: http://math.stackexchange.com/questions/37255/galois-theory-finding-roots-of-a-polynomial and http://math.stackexchange.com/questions/117616/when-can-galois-theory-actually-help-you-find-the-roots-of-a-polynomial Please take a look at them, and remember that your original question is about you wanting a simple application of Group Theory. – Martin Argerami May 12 '12 at 02:27
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    @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 07:36