= 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 $f=\operatorname{ord}_8(p)$. 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 $p\equiv1\pmod8$ and $f=2$ for the other three odd residue classes.
At $p=2$, the translated <cyclotomic polynomial> is
$$
\Phi_8(1+T)=(1+T)^4+1=T^4+4T^3+6T^2+4T+2.
$$
It 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
$$
\boxed{\operatorname{Gal}(\mathbb Q_p(\zeta_8)/\mathbb Q_p)\cong
\begin{cases}C_2\times C_2,&p=2,\\1,&p\equiv1\pmod8,\\C_2,&p\equiv3,5,7\pmod8.\end{cases}}
$$
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