3
$\begingroup$

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.

  • 1
    @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.2013-10-07

1 Answers 1

7

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.

  • 13
    @LuizRobertoMeier Yes, "Galois theory can be used to solve polynomials o$f$ any de$g$ree", 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.2012-05-13