Compute the addition table of a field with $8$ elements. Hint: Factor $x^8-x$ over $\mathbb{Z_2}$.
In the previous theorem, the book shows a proof of the existence of a field having $p^n$ elements by using the smallest field over which $x^q-x$ splits, which is what it is hinting at.
What I did is to add the seven roots of unity $\zeta=e^{i2\pi/7}$ to $\mathbb Z_2$. However, to fully express the table, I need to know what element corresponds to the expression $1+\zeta$, and then all follows, but I can't compute it.
I believe you will have an easier time using Daniel Schepler's hint:
Let $p(x) = x^3 + x + 1 \in \Bbb Z_2[x]$, with $u$ a root. I leave it to you to show this is irreducible over $\Bbb Z_2$.
Clearly, $u^3 = u+1$. The powers of $u$ are:
$u^2$ (we cannot reduce this further),
$u^3 = u+1$, $u^4 = u(u^3) = u(u+1) = u^2 + u$ (again, this cannot be further reduced).
$u^5 = u^2(u^3) = u^2(u+1) = u^3 + u^2 = u^2 + u + 1$,
$u^6 = (u^3)^2 = (u+1)^2 = u^2+1$,
and, of course, $u^7 = u(u^6) = u(u^2+1) = u^3 + u = (u+1) + u = 1$.
The problem with your approach straight off the bat is: "What is $\zeta$ ?", since the expression $e^{i2\pi/7}$ makes no sense over the field $\Bbb Z_2$.
Clearly, $\Bbb Z_2(u) \cong \Bbb Z_2[x]/(p(x))$ is a field of eight elements. Addition is straight-forward, as we have:
$\Bbb Z_2(u) = \{a_0 + a_1u + a_2u^2\mid a_0,a_1,a_2 \in \Bbb Z_2\} \cong (\Bbb Z_2)^3$ as a $\Bbb Z_2$ vector-space.