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How can I show that Trace $(ax) = 0$ implies that $a = 0$ in a field $F$ (over F2) of order $2^N$? I get that I can something like the following:

Trace($ax_1)=$ Trace($ax_2$) $\implies$ $a(x_1-x_2)+a^2(x_1-x_2)^2+...+a^{2^{N-1}}(x_1-x_2)^{2^{N-1}} = 0,$

but I don't see where to go from here or why this is useful.

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