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Thread: automorphisms

  1. #1
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    automorphisms

    There are Q(rational)-automorphisms $\displaystyle \sigma$ and $\displaystyle \tau$ of L s.t

    $\displaystyle \sigma(\epsilon) = \epsilon^2$
    $\displaystyle \sigma(\alpha) = \alpha$
    $\displaystyle \tau(\epsilon) = \epsilon$
    $\displaystyle \tau(\alpha) = \epsilon*\alpha$

    can anyone explain to me why the above 4 are enough to determine the effect of [$\displaystyle \sigma$ and $\displaystyle \tau$ on the whole of L

    thanks
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  2. #2
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    From your previous posts, $\displaystyle L=\mathbb{Q}(\alpha,\epsilon)=\left\{\displaystyle \sum_{i=0}^4 \sum_{j=0}^3 a_{ij} \alpha^i\epsilon^j: a_{ij}\in\mathbb{Q}\right\}$, and any automorphism $\displaystyle f$ fixing $\displaystyle \mathbb{Q}$ satisfies $\displaystyle f(\sum_{i=0}^4 \sum_{j=0}^3 a_{ij} \alpha^i\epsilon^j)=\sum_{i=0}^4 \sum_{j=0}^3 a_{ij} f(\alpha)^if(\epsilon)^j$. Therefore if we know the images of $\displaystyle \alpha $ and $\displaystyle \epsilon$, we completely determine the map, i.e. every other image can be calculated by knowing these images.
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