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Math Help - Decomposition field over Q

  1. #1
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    Decomposition field over Q

    Let f = X^4 + X^2 + 1 \in \mathbb{Q}[X], and \omega = \frac{1}{2}(-1+i\sqrt{3}) \in \mathbb{C}
    Prove that \mathbb{Q}(\omega) is a decomposition field of f over \mathbb{Q}.


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  2. #2
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    X^{4}+X^{2}+1=(X^{2}+X+1)(X^2-X+1)
    \omega and {\omega}^2 are the roots of (X^{2}+X+1). The roots of (X^2-X+1) are \omega ' and {\omega '}^2 where \omega '=\frac{1}{2}(1-i\sqrt3)
    As \mathbb{Q}(\omega ) contains \omega, \omega, {\omega}^2, {\omega '}^2, it is a decomposition field of f over \mathbb{Q}
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