In $\mathbb{Z_2[x]}/(x^2 + x +1)$, what is $\overline{x} \cdot \overline{x}$? In $\mathbb{Z_2[x]}/(x^2 + x +1)$, what is $\overline{x} \cdot \overline{x}$?
$\overline{x} \cdot \overline{x} = \overline{x^2}$
If I divide $\overline{x^2}$ by $(x^2 + x +1)$ I get an answer of $1$ with remainder $(-x-1)$.
$\overline{-1}$ in $\mathbb{Z_2}$ is $\overline{1}$
and
$\overline{-x}$ in $\mathbb{Z_2}$ is $\overline{x}$
So $\overline{x} \cdot \overline{x} = \overline{x + 1}$
Have I got that right?

Observe that $$\overline{x^2 + x + 1} = \overline 0$$ since $x^2 + x + 1 \in (x^2 + x + 1)$, which means that $$\overline{x^2} =\overline{ - (x + 1)},$$ but since $-1 \equiv 1 \pmod 2,$ we get $$\overline{x^2} = \overline{x + 1}.$$
