This is my attempt to find $\vert E \vert$, which is the order of the field $E$. If I am on the wrong track, please guide me to a technique that will work with more general fields and polynomials.
We have a field $F$ with $4$ elements (which we can denote by $0,1,1+1,1+1+1)$. Factorization: $$x^8-1=(x^4+1)(x^4-1)=(x^2+1)(x^2-1)(x^4+1)\\=(x+1)(x-1)(x^2+1)(x^4+1)$$ so if we adjoin $i$ to $F$ ($x^2+1$ will factor) we have $$x^8-1=(x+1)(x-1)(x+i)(x-i)(x^4+1)$$ and if we adjoin $\sqrt 2$ then $x^4+1$ will factor into $(x^2 + \sqrt 2 x+1)(x^2 - \sqrt 2 x + 1)$. So $E=F(i,\sqrt 2)$. Thus, $E$ has $4^3=64$ elements.
Is my work correct? In general, if I don't know the elements of my field, how can I determine whether new elements have to be added into my field in order to factorize? In this case, how do we know $i, \sqrt 2\not \in F$?
Since the characteristic is $2$, we have$$x^8-1=(x^4-1)^2=(x^2-1)^4=(x-1)^8.$$This should make everything a lot easier.