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Let $\xi\in GR(4^{m})$ and $\xi$ is of order $2^{m}-1$. Let $a=l-k$ for l and k are distinct in the range $[0,2^{m}-2]$, where $m\geq 2$. Why $2\xi^{a}=0$ is a contradiction? Could you help me to explain the reasons?

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    What is $GR(4^m)$?2012-07-25
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    Galois rings with $4^{m}$ elements is denoted by $GR(4^{m})$2012-07-27
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    $GR(4^{m})=\{a_{0}+a_{1}x+a_{2}x^{2}+...+a_{m-1}x^{m-1}|a_{i}\in \mathbb{Z}_{4},i=0,1,2,...,m-1\}$2012-07-27

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