let be $g=x^5+x+1\in\mathbb{Z}_2[x]$, find splitting field $F$ of $g$ over $\mathbb{Z}_2$, determine $\text{Gal}(F/\mathbb{Z}_2)$ and describe a lattice of all subfields of $F$.
One can check, that $g=(x^3+x^2+1)(x^2+x+1)$. The first polynomial splits completely in $\mathbb{Z}_2[x]/(x^3+x^2+1)=\mathbb{F}_8$, similarly the second in $\mathbb{Z}_2[x]/(x^2+x+1)=\mathbb{F}_4$. Both of these are contained in $\mathbb{F}_{64}=F$. From this, I also have a complete lattice of subfields consisting of $\mathbb{Z}_2,\mathbb{F}_4,\mathbb{F}_8$ and $F$. But how can I compute degrees $[F:\mathbb{F}_4], [\mathbb{F}_8:\mathbb{Z}_2],$ etc.?
And finally, I have no idea, how to find Galois group...
Thx for help.
Your fields are finite sets, so the degree shouldn't be so hard if you think about it for a few seconds. The Galois group of an extension of finite fields is always cyclic, generated by the Frobenius automorphism. So once you have the degrees, there is also nothing more to do here.