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Math Help - Galois Theory

  1. #1
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    Galois Theory

    How can I find the Galois group of x^5 + 2x +2?
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    Quote Originally Posted by peteryellow View Post
    How can I find the Galois group of x^5 + 2x +2?
    If you reduce mod 29 the polynomial factors as (x^2+4x+7)(x^3+25x^2+9x+21). Sorry if this method is really computational, this is the only method I was able to think of.

    Therefore by Dedekind's theorem the Galois group contains a product of a 2-cycle and 3-cycle. The only transitive subgroup of S_5 which contains a product of a 2-cycle and a 3-cycle is S_5 itself. Therefore, the Galois group is S_5.
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    Thanks, but I have a question why are you reducing modulo 29 why not any other number?

    I have also seen this method somewhere else, but I couldnot see why they choose these numbers as you have choosen 29?
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    Quote Originally Posted by peteryellow View Post
    Thanks, but I have a question why are you reducing modulo 29 why not any other number?

    I have also seen this method somewhere else, but I couldnot see why they choose these numbers as you have choosen 29?
    I started from p=3 and worked myself up to p=29 until I got a the result I wanted.
    I used computer software for this, I did not spend all my time doing this by hand.
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