 Solution

For odd $p$, eighth roots of unity have order prime to the residue characteristic. The total ramification criterion for local cyclotomic extensions shows that they generate an unramified extension of degree $f={\mathrm{ord}}_8(p)$. More directly, a primitive eighth root in ${\mathbb{F}}_{p^f}$ lifts uniquely by Hensel lemma, since the derivative of $T^8-1$ is a unit at each nonzero root. Its residue has exact order eight, and the Frobenius automorphism acts by ${\zeta }_8\mapsto {\zeta }_8^p$. Consequently $f=1$ when $p\equiv 1(\mathrm{mod}8)$ and $f=2$ for the other three odd residue classes.

At $p=2$, the translated cyclotomic polynomial isIt is Eisenstein at two, so the extension has degree four. All primitive eighth roots are odd powers of ${\zeta }_8$ and already lie in the field; the extension is Galois, and its automorphisms give the full group $(\mathbb{Z}/8\mathbb{Z}{)}^{\times }\cong C_2\times C_2$. Thus