Let's consider the polynomial $m(x)$ over a field $\mathbb{Z}_{3}$. We know that $[m(x)]_{m(x)}=m(a)=0$.
Now $m(x)=x^{3}+1$; in my lectures slide, it's said at this point that:
$m(a)=0$ implies that $a^{3}=2$.
What this means ?
Moreover, in the slides about the modulo arithmetic with polynomials it's said that, when we work with polynomial whose coefficients are modulo n then the negative coefficients equals to the positive ones. But why ?
We are working in modulo 3, so if $m(a) = 0$, we have $a^3 \equiv -1 \equiv 2 \pmod 3$. So this means $a^3 = 2$.