My Problem is to consider if the polynomial $x^2 + 1$ is irreducible over $\mathbb{Z}_{/3}[x]$
My Approach was: after looking closer onto the given Facts, i can see that $\mathbb{Z}/_{3}[x]$ is a polynomial ring that consists of: $$\mathbb{Z}/_{3}[x]=\{ 0, 1, 2, x, x+1, x+2, 2x, 2x+1, 2x+2 \} $$ On the other Hand, the given polynomial can be transformed into: $$x^2 + 1 = (x+1)\cdot (x+1)$$ And due to $(x+1)\in \mathbb{Z}/_{3}[x]$ i can state that the given polynomial is actually not irreducible.
My question is: am i right? And in case i am wrong, what are my mistakes?
Post scriptum: as i was told in the comments, i made some mistakes. $\sqrt{x^2 +1}$ has no root in the given ring, so the polynomial is in fact irreducible.
Hint: Test if $x^2+1$ vanishes for some $x \in \mathbb F_3$. If it does, then it is a product of linear factors. If not, then it cannot be reducible.