Evaluating products in a Galois field. I have the Galois field $$GF(27)$$
I know that $$x^3+2x+1$$ is irreducible in $$Z_3$$
It follows that $$GF(27)=Z_3[x]/<f>$$
How would I evaluate $$\alpha^2(\alpha^2+\alpha+1)$$ in my Galois field?

If for example you are working over the isomorphic copy $GF(27)=GF(3)[x]/(x^3+x^2+x+1)$ You know that $\alpha^3+\alpha^2+\alpha+1=0$, so your expression would be $$\alpha^3(\alpha^2+\alpha+1)=2(\alpha^2+\alpha+1)(\alpha^2+\alpha+1)=2(\alpha^4+\alpha^2+1+2\alpha^3+2\alpha^2+2\alpha)=2(\alpha^4+\alpha^2+1)=2(\alpha \alpha^3+\alpha^2+1)=2(\alpha(2(\alpha^2+\alpha+1))+\alpha^2+1)=2(2\alpha^3+2\alpha^2+2\alpha+\alpha^2+1)=2(\alpha^2+2)=2\alpha^2+1$$
