I have to use the polynomial $X^8 + X^4 + X^3 + X^2 +1$, and given $\alpha$ is a primitive element. So far I have that $\alpha^8=\alpha^4 + \alpha^3+\alpha^2+1$, and the only method for finding the elements of the field known to me is to multiply with $\alpha$ and replace $\alpha^8$ with $\alpha^4 + \alpha^3+\alpha^2+1$. For example, the first 9 elements of the field are $0, 1, \alpha, \alpha^2, ...,\alpha^7$. Then I find the others as
$\alpha^8=\alpha^4 + \alpha^3+\alpha^2+1$,
$\alpha^9= \alpha^5 + \alpha^4 + \alpha^3+ \alpha.$
So on...
My problem with following this system is that I make too many mistakes. Is there another way to construct this field, or is there a pattern that the elements of the field follow?