automorphisms

**dopi** · March 24th 2009, 09:47 AM

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

**siclar** · March 24th 2009, 12:30 PM

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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