Is $ x^3-x^2 -1 $ reducible in $\mathbb Z_5[x]/(x^3-x^2-1) = \mathbb Z_5(u) $?

Question

Let $F=\mathbb Z_5[x]/(x^3-x^2-1)=\mathbb Z_5(u)$, $u=[x]$.

a) Is $ p(x) = x^3-x^2 -1 $ reducible over $F$? We found that $p(x)$ is irreducible in $\mathbb Z_5[x]$, but we are not sure about $F$.

b) We found that a basis for $F$ over $\mathbb Z_5$ is $\{1,u,u^2\}$ and we want to express $\frac{1}{1+u}$ in this basis.

Answer 1

a)
$p(x) = 0$ (or, more rigorously, $[p(x)] = [0]$) in $F$. Is that reducible?

b)
I think the easiest way to do that is to use the extended Euclidean algorithm in $\Bbb Z_5[x]$ to find polynomials $f, g$ such that $$ f(x)(1+x) + g(x)(x^3 - x^2-1) = 1 $$ Then projecting this equality down to $F$ we find $f(u)(1+u) = 1$, which means that $f(u)$ is exactly what you're looking for.