Polynomial irreducibility over $\mathbb{Z}_{3}[x]$

Question

Let $f(x) = x^4 + x^2 + x + 1 \in \mathbb{Z}_{3}[x]$

Show that $f$ is irreducible over $\mathbb{Z}_{3}$, then factor $f$ over $K = \frac{\mathbb{Z}_{3}[x]}{(f(x))}$

I first tried to use Eisenstein criterion to prove the irreducibility but since we are working over $\mathbb{Z}_{3}[x]$, neither $f(x)$, $f(x+1)$ nor $f(x+2)$ satisfies the Eisenstein criterion and i can't find another way to prove this irreducibility.

Any help would be truly appreciated! Thanks.

Answer 1

Maybe not the quickest way but a method nonetheless:

If it was to be reducible, it must either have a linear factor or a quadratic factor.

A quick check shows it cannot be linear so it has to be $$f(x) = P1(x)P2(x)$$ where both $P1$ and $P2$ are irreducible quadratics.

Also note that it has to be the case that either both $P1(x)$ and $P2(x)$ have leading coefficient of $1$ or both have the leading coefficient $2$.

Now a quick search shows there are $3$ irreducible quadratics with leading coeff $1$ and similarly for leading coeff $2$. So just by checking $6$ possibilities should give you the irreducibility.