Find the multiplicative inverse in the given field.
$\displaystyle [x^2-2x+1]$ in $\displaystyle Z_3[x]/<x^3+x^2+2x+1>$
Thank you!
Hint: $\displaystyle \mathbb{Z}_{3}[x]{/}(x^3 + x^2 +2x + 1) $ = $\displaystyle \{0,1,2,x,x+1,x+2,2x,2x+1,2x+2,x^2,x^{2} +1, x^{2}+2,x^{2}+x,x^{2}+2x,x^{2}+x+1,$$\displaystyle x^{2}+x+2,x^{2}+2x+1,x^{2}+2x+2\}$
And $\displaystyle x^2 -2x +1 = x^2 +x + 1 $