Different Representations of GF(8) Can anyone point me in the right direction with the following problem?
Given that 
$$GF(8)=\frac{Z_{2}}{x^3+x^2+1}= \frac{Z_{2}}{x^3+x+1}$$
Find $\beta$ as a function of $\alpha$ , where $\alpha$ is a root of $x^3+x+1$ and $\beta$ is a root of $x^3+x^2+1$

Let $\alpha$ be a root of $x^3+x+1$.  Then a general element of $GF(8)$ has the form $a+b\alpha + c\alpha^2$ for some $a,b,c \in GF(2)$.  Need to solve for for $a,b,c$ in $(a+b\alpha + c\alpha^2)^3+(a+b\alpha + c\alpha^2)^2+1 = 0$  Recall that $u^2=u$ for any $u \in GF(2)$ and $2u=0$ for any $u \in GF(8)$.  After simplifying you'll find 
$$ (1+b+c+bc) + (b+c+ab+ac)\alpha + (b+ab)\alpha^2 = 0$$
from which you can determine that $\beta$ can be taken to be $1+\alpha, 1+\alpha^2, $ or $1+\alpha+\alpha^2$
