Factoring polynomials: $p_2(t)=2t^4+t^2+2 \in \mathbb{Z}_3[t]$

Question

I want to factor the following polynomials into irreducible polynomials: $$p_1(t)=t^4+1 \in \mathbb{Z}_3[t]$$

$$p_2(t)=2t^4+t^2+2 \in \mathbb{Z}_3[t]$$

$\mathbb{Z}_3:$

$\begin{pmatrix}+_{\mathbb{Z}_3} & 0& 1 &2 \\0 &0 &1 &2 \\1 &1 &2 &0 \\2 &2 &0 &1\end{pmatrix}$

$\begin{pmatrix}\cdot_{\mathbb{Z}_3} & 0& 1 &2 \\0 &0 &0 &0 \\1 &0 &1 &2 \\2 &0 &2 &1\end{pmatrix}$

So if I'm not wrong both polynomials should have no roots so I think there is nothing to do here. But according to my professor there is something to be done about $p_2$ and I could use some help there.

Thanks in advance!

Answer 1

In $\mathbb{Z}_3[t]$, you have$$t^4+1=(t^2-1)^2+2t^2=(t^2-1)^2-t^2=(t^2-t-1)(t^2+t-1)$$and$$2t^4+t^2+2=-t^4+t^2-1=t^2-\bigl((t^2+1)^2-2t^2\bigr)=-(t^2+1)^2.$$