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Math Help - automorphisms

  1. #1
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    automorphisms

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

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

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

    thanks
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  2. #2
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    From your previous posts, 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 f fixing \mathbb{Q} satisfies 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 \alpha and \epsilon, we completely determine the map, i.e. every other image can be calculated by knowing these images.
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