A property of finite field of order $2^n$ 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 $a=b$,
then $a^2+ab+b^2=3a^2=a^2=0$ and $a=0=b$, 
we are done.
Suppose that $a\neq b$.
Observe that $0=(a-b)(a^2+ab+b^2)=a^3-b^3$.
Thus, $a^3=b^3$.
We claim that $a=0$ and $b=0$.
If $a\neq 0$,
then $(a^{-1}b)^3=1$ and the multiplicative order of $a^{-1}b$ 
in the multiplicative group $F-\{0\}$ is $1$
because $3\not\mid |F-\{0\}|=2^n-1$.
Hence, $a^{-1}b=1$ and $a=b$,
a contradiction.
Therefore, $a=0$.
By a similar argument, we have $b=0$.
