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Thread: Decomposition field over Q

  1. #1
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    Decomposition field over Q

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


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