Given a polynomial: $f(x) = x^2 + 2x + 2$ over $GF(3)$. I want to know if i can use it to construct $GF(3^2)$.
- This equation satisfies first condition: A primitive polynomial is irreducible.
- The second condition i'm trying to confirm is that i can use it generate $GF(3^2)$.
$\alpha^2 + 2\alpha + 2 \rightarrow \alpha^2 = \alpha + 1$.
$\alpha^1 = \alpha$
$\alpha^2 = \alpha+1$
$\alpha^3 = \alpha*\alpha^2 \rightarrow1 + 2\alpha $
$\alpha^4 = 2 $
$\alpha^5 = 2\alpha$
$\alpha^6 = \alpha+2$
$\alpha^7 = 1$
Now Here $\alpha^8$ should be equal to $1$ instead of $\alpha^7 .$
$\alpha^8 = \alpha$
What i am doing wrong , any help would be great.