Consider the extension field $F_{q^n}$ as a vector space over $F_q$, find two linearly independent elements $\alpha_1$ and $\alpha_2$ in $F_{q^n}$, such that $\alpha_{i}\neq1$, for $i=1,2$, and : $$\alpha_1^2\alpha_2^8+\alpha_1^8\alpha_2^2+\alpha_1\alpha_2^8+\alpha_2\alpha_1^8+\alpha_1\alpha_2^2+\alpha_2\alpha_1^2=0$$
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Hmm. A curious question. I can think of a context where a similar question might arise, so I want to hear about yours. Please share! Mind you, the question is a bit unclear in the sense that which parameters ($q$, $n$), if any, are fixed, and which, if any, can be chosen by the answerer. Also, I am a bit tempted to restrict myself to the case of characteristic two, as it is easier to think of a nice context then :-) – Jyrki Lahtonen Aug 20 '14 at 06:37
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For example, if $q=2,n=3$, then any pair linearly independent etc pair $(\alpha_1,\alpha_2)$ will work. Trivially. – Jyrki Lahtonen Aug 20 '14 at 06:39
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Thanks Jyrki, well Actually I need the answer for $6\leq n$. If there is an answer even for characteristic $2$ it is okay, – user27932 Aug 20 '14 at 06:45