Let $(K,v)$ a complete discrete valuation field of mixed characteristics $(0,p)$ and let $\bar{K}$ ann algebraic closure of $K$.
We can define an additive valuation $w: \bar{K} \to \mathbb{Q}\cup \{\infty\}$ on $\bar{K}$ through the intermediate field. If $x \in \bar{K}$ and $L$ is an intermediate field ($K \subseteq L \subseteq \bar{K}$) such that $x \in L$, then $$w(x)=\frac{1}{n}v(N_{L\K}(x)).$$ For me it’s clear that it is an additive valuation and that $w|_K=v$.
Now my claim is to prove that if we have $K \subseteq K’ \subseteq \bar{K}$ $$K’/K \text{ is unramified} \iff \text{for every finite extension $E$ of $K$ with $E \subseteq K’$, we have $E/K$ is unramified} $$
A field extension $E/K$ is unramified if $w(E)=\mathbb Z$. Thus, we hope to check that for a field extension $K'/K$, we have $w(K')=\mathbb Z$ if and only if $w(E)=\mathbb Z$ for all finite subextensions $K'/E/K$. But this is clear since $K'=\bigcup_{K'/E/K}E$, which is a union over all finite extensions $E/K$, so $$w(K')=\bigcup_{K'/E/K}w(E).$$ Thus, if $w(E)=\mathbb Z$ for all $K'/E/K$ then $w(K')=\mathbb Z$. Conversely, if $w(K')=\mathbb Z$ then for all finite $K'/E/K$ we have $w(K)=\mathbb Z\subseteq w(E)\subseteq w(K')=\mathbb Z$, i.e., $w(E)=\mathbb Z$.