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Math Help - Galois Group

  1. #1
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    Galois Group

    Find the galois group of f(x) =1+x+x^2/2+x^3/6+x^4/24.

    I multiply the polynomial with 24 then get
    24f(x) =24+24x+12x^2+4x^3+x^4.

    Then I have that the cubic resolvent of 24f(x) is

    h(x) = z^3-12z^2+192.
    Therefore, I get that the galois group of 24f(x) is A_4.

    My question is: Can I so conclude that the Galois group of f(x)
    is A_4.
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  2. #2
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    Quote Originally Posted by ZetaX View Post
    Find the galois group of f(x) =1+x+x^2/2+x^3/6+x^4/24.

    I multiply the polynomial with 24 then get
    24f(x) =24+24x+12x^2+4x^3+x^4.

    Then I have that the cubic resolvent of 24f(x) is

    h(x) = z^3-12z^2+192.
    Therefore, I get that the galois group of 24f(x) is A_4.

    My question is: Can I so conclude that the Galois group of f(x)
    is A_4.
    Let a\in \mathbb{Q}^{\times} then f(x) and af(x) have the same Galois group. Because the splitting field of f(x) and af(x) are identical. The Galois group of a polynomial is defined to be the Galois group of the splitting field over the base field, since the splitting fields are the same it follows that the Galois groups are the same.
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