Let $f = x^3 + x + 1$. How many elements does $\mathbb{Z}_2[x]/(f)$ have? Write out all of the elements of this field, and find the inverse of each nonzero element.
I know that $f$ is irreducible in $\mathbb{Z}_2[x]$ and thus has no roots. How would I calculate what kinds of elements are contained in $\mathbb{Z}_2[x]/(f)$? What's the form of elements in $\mathbb{Z}_2[x]/(f)$?
Each element of $\Bbb Z_2[x]/(f)$ has the form $a_0+a_1x+a_2x^2+(f)$ where $a_0,a_1,a_2\in\Bbb Z_2$. There are two possibilities for each $a_i$ so $2\times2\times2$ in all.