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Math Help - Galois group

  1. #1
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    Galois group

    Find the roots of f(x)=x^4+x^2+1. Find the Galois group of f(x).
    I know that I need to use the resolvent cubic, but I donít know how to find it and use it. Can someone explain it to me? Thanks.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by apple2009 View Post
    Find the roots of f(x)=x^4+x^2+1. Find the Galois group of f(x).
    I know that I need to use the resolvent cubic, but I donít know how to find it and use it. Can someone explain it to me? Thanks.
     x^4+x^2+1=(x^2+x+1)(x^2-x+1)

    Can you take it from here?
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  3. #3
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    Quote Originally Posted by apple2009 View Post
    Find the roots of f(x)=x^4+x^2+1. Find the Galois group of f(x).
    I know that I need to use the resolvent cubic, but I donít know how to find it and use it. Can someone explain it to me? Thanks.
    Let a_3=0, a_2=1, a_1=0, and a_0=1. Plug them into the polynomial for a resolvent cubic (see here)

    If my computation is correct, the resolvent cubic of f(x) is x^3-x^2-4x+4.

    Then check how the resolvent cubic of f(x) factors and find the corresponding galois group (see here)
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