Find a polynomial of degree 5 that is irreducible over $\mathbb{Z}_3$. Can someone please let me know if this looks ok? Thanks in advance!
An irreducible degree $5$ polynomial over $\mathbb{Z}_3$is one such that $f(0)\neq0,f(1)\neq0,f(2)\neq0$.
Take e.g. $p(x)=x^5+x^4+x^3+x^2+1$ 
$p(0)=1$
$p(1)=2$
$p(2)=1$
And,  $$x^2,x^2+1, x^2+x+1\ \nmid\ x^5+x^4+x^3+x^2+1.$$
$\therefore x^5+x^4+x^3+x^2+1$ is an irreducible polynomial over $\mathbb{Z}_3$.

Since it has degree $5$, it would need to have a root ( but it doesn't), or be divisible by an irreducible quadratic.
But, did you check $x^2+2x+2$?
There may be others.
