My problem is
Find the splitting field, $F$, of $x^5+x+1\in\mathbb{Z}_2[x]$ and determine the degree $[F:\mathbb{Z}_2]$.
I was able to factor $x^5+x+1=\left(x^2+x+1\right)\left(x^3+x^2+1\right)$ and both of the factors are irreducible over $\mathbb{Z}_2[x]$, I think.
Since $\mathbb{Z}_2$ is a field and $x^2+x+1\in\mathbb{Z}_2[x]$ is irreducible and monic, then $\mathbb{Z}_2[x]\,/\left(x^2+x+1\right)$ is a field extesion of $\mathbb{Z}_2$. Then $c:=x+\left(x^2+x+1\right)$ is a root of $x^2+x+1$ (in which field?). Similary with the other factor. But how is this helpful?
Each of the factors is irreducible because they are degree at most $3$ with no roots. Now we know that there is a root of the quadratic factor in the (unique) degree $2$ extension of $\Bbb F_2$, and similarly for the cubic factor in the degree $3$ extension. Now each of these fields is contained in the degree $6$ extension, and clearly it is the smallest degree extension which has subfields of degrees $2$ and $3$, so the splitting field is $\Bbb F_{2^6}$, the field with $2^6$ elements.