Let $F$ = $(\mathbb{Z}/3\mathbb Z)[x]/(x^2-2)$ be the quotient ring generated by the principal ideal $(x^2-2)$. I need to find a root of $x^2+2x+2$ and find all generators of the group $F^*$.
Is $x+1$ a root for the above? I have that $(x+1)(2x) = 2x^2 + 2x = x^2 + x^2 + 2x + 2 - 2 = x^2 + 2x +2$. Is this valid?
As for finding all generators, I am a bit stuck. I could try all possibilities (like $x+1$), but aren't there $3^3 = 27$ cases?