<s>[INST] Finding a pair of elements to satisfy an inequation Let $F$ be a field of characteristic 2 with more than 2 elements. Show that there are elements $a$ and $b$ in $F$ such that $(a+b)^3 \not= a^3 + b^3$.

$F$ couldn't possibly have less than 2 elements, and if it had exactly 2 — that is, $F = \mathbb Z_2$ —, $(a+b)^3$ would actually always be $a^3+b^3$. With $\#F>2$, how do I find $a$ and $b$ to violate that?
All I could do so far is simplify the inequation, using the characteristic and the commutative prroperty:
$$\begin{align}
(a+b)^3 &\not= a^3 + b^3\\
a^3 + 3a^2b + 3ab^2 + b^3 &\not= a^3 + b^3\\
a^3 + a^2b + ab^2 + b^3 &\not= a^3 + b^3\\
a^2b + ab^2 &\not= 0\\
ab(a + b) &\not= 0\quad.
\end{align}$$
 [/INST] In a field, product is $0$ iff atleast one of the element in product is $0$. Here, if $a\neq0,b\neq 0$ and $a\neq -b$ then , $ab(a+b)\neq 0$, so choose non-zero elements of field $\Bbb F$ such that one is not the additive inverse of other one.
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