Expanding an expression in a certain field If $\mathbb F_2$ is a field of characteristic $2$, then we have $x+x=y+y=z+z=0$ for all $x,y,z \in \mathbb F_2$. When I expand $(x+y)(y+z)(z+x)$, I get
\begin{align}
(x+y)(y+z)(z+x) &= xz^2+y^2z+yz^2+x^2y+x^2z+xy^2+2xyz \\ 
&= xz^2+y^2z+yz^2+x^2y+x^2z+xy^2+(x+x)yz \\ 
&= xz^2+y^2z+yz^2+x^2y+x^2z+xy^2.
\end{align}
I got $2xyz=0$ in $\mathbb F_2$, but how can I simplify my above expression even more?

In $\mathbb{F}_2$, we also have $x^2=x$ and so on. So your expression will result in $0$. Another way to look at it is that at least two of $x,y$ or $z$ will take the same value, in which case at least one of $x+y, y+z, x+z$ will be $0$.
