Suppose $a$ and $b$ are elements of a finite field of order $2^n$ with $n$ odd and $a^2+ab+b^2=0$. Is it necessary that both $a$ and $b$ must be zero ?
I understand that the field has characteristic $2$ but don't know how to use the fact that $n$ is odd, please help.
If $b$ were not 0 then $a/b$ would be a root of $x^2 + x + 1$, which is irreducible over ${\mathbf F}_2$. Look at the size of the field ${\mathbf F}_2(a/b)$ and the size of the field you are working in that has order $2^n$.