Show that (x+y)^3 is not equal to x^3+y^3 for some x and y in a field F

Question

Problem

Let F be a finite field of characteristic 2 with more than two elements. Show that $(x+y)^3 \neq x^3+y^3$ for some $x,y \in F$

Doubt

If $2x=0$ for all $x \in F$ ,then

$(x+y)^3=x^3+y^3+3xxy+3yyx =x^3+y^3+xxy+yyx$

Any suggestion or hint what to do after this .

Answer 1

First, note that $$ x^3 + y^3 = (x + y)^3 \iff 0 = xy(x + y). $$ So, you need to choose $x$ and $y$ such that both $x$ and $y$ are nonzero, and $x + y\neq 0.$ Can you show that you can always choose such elements in a characteristic two field $F\not\cong\Bbb F_2$? Hint: Start with $x = 1.$ What can't $y$ be?

Answer 2

Well, you are going to have to consider a field extension. The polynomial $t^2+t+1\in\mathbb{F}_2[t]$ is irreducible. What equations can you try in $F=\mathbb{F}_2[t]/(t^2+t+1)$?