Let $F$ be a number field and let $\mathfrak{p}\in \mathsf{Spec} \: \mathcal{O}_F$. We have a non Archimedean valuation $\nu_\mathfrak{p}\colon F\longrightarrow\mathbb{R}_{\geq 0}$, given by $\nu_p(x):=\mathsf{card}(\mathcal{O}_F/\mathfrak{p})^{\mathsf{ord}_\mathfrak{p}(x)}$. $F_\mathfrak{p}$ is completion with respect to valuation $\nu_p(x)$.
If $F$ and $K$ are number field and $F/K$ is finite galois extension.
My question is,
Does $F_\mathfrak{p}/K_\mathfrak{p}$ is totally ramified implies $\mathfrak{p}$ is totally ramified in $F/K$ ?
I think this kind of statement holds if we add (although may be we need to add some condition). Reference(pdf, book, etc) is also appreciated.
Try with $\Bbb{Q}(\zeta_3,7^{1/3})/\Bbb{Q}$ and $\mathfrak{p} = (\zeta_3-2,7^{1/3})$