# Thread: Multiplicative Inverse

1. ## Multiplicative Inverse

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!

2. 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 $\displaystyle x^2 -2x +1 = x^2 +x + 1$

3. I think I've got it!
Is the answer (x-2) (or should I say x+1?)?

Correct me if I've done this wrong, but:
x^2-2x+1)(x-2)=x^3-2x^2+x-2x^2+4x-2
=x^3-4x^2+5x-2
=x^3+x^2+2x+1

4. Actually, it seems I am going about this the wrong way.
I am trying the find an a^-1 in which:
a*a^-1=1 in Z_3.

where a=x^2+x+1
Is this right? By doing a case by case basis I still cant find a suitable answer.

5. I'm not sure if I'm allowed to bump this, but it would be nice to have an answer.
Thank you!