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Math Help - Galois group of a polynomial

  1. #1
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    Galois group of a polynomial

    The problem is to identify Galois group of the polynomial f=x^4 + 2x^2 + x +3.
    We have done a couple of examples in class, but the I am unable to solve this.

    Thank you.
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  2. #2
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    There are many ways to do it. Which method have you learned so far?
    You can use reduction modulo p, or you can use the method where you find cubic resolvent and then determine the Galois group. Have you learned this?

    Anyways, It is easily seen that the polynomial is irreducible over \mathbb Q. You can try to show this as following.

    It is easily seen by Rational root theorem that the polynomial has no rational root, furthermore assume that the polynomial can be written as
    x^4+2x^2+x+3 = (x^2+bx+c)(x^2+dx+e) and then try to get a contradiction. Therefore you can conlude that the polynomial is irreducible over \mathbb Q.

    By reducing the polynomial modulo 3 we have  (x^3+2x+1)(x) and modulo 5 we have
    (x+2)(x+1)(x^2+2x+4). Since The Galois group contains a trasoposition and a 3-cycle we conclude that the Galois group is S_4.
    Last edited by peteryellow; April 19th 2009 at 05:45 AM.
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  3. #3
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    Quote Originally Posted by peteryellow View Post
    There are many ways to do it. Which method have you learned so far?
    You can use reduction modulo p, or you can use the method where you find cubic resolvent and then determine teh Galois group. Have you learned this?
    Yes, we've learned the method with cubic resolvent, but I'm unable to solve it by myself.
    I would be really grateful for your help and time.
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  4. #4
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    The cubic resolvent is
     g(y) = y^3-2y^2-12y-25 (If I have solved it correctly.)You can recheck it. then it is easily seen that the Galois group if g is S_3.
    Then by using some Theorems you can conclude that S_4 is the Galois group.

    Ask if something is not very clear.
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  5. #5
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    Thank you very very very much, I will read more about Galois groups and than ask if there are any questions left.
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