Let $K \supset k,$ let $\mathfrak{P}$ be a prime ideal in $K$ and let p be the prime ideal of $k$ divisible by $\mathfrak{P}$ . Show that $\mathfrak{P}$ is wildly ramified if and only if $\operatorname{Tr}_{K_{\mathfrak{P}} / k_{\mathfrak{p}}} \alpha$ is in $\mathfrak{p}_{\mathfrak{p}}$ for every $\alpha$ in $\mathfrak{O}_{\mathfrak{P}}$.
I think I can use the proposition $\mathfrak{P}^{e} | \mathfrak{d}_{K / k}$ if and only if $\mathfrak{p} | e$, but I have no idea what to do then.
Abbreviate $\mathfrak{p}_\mathfrak{p}$ as $\mathfrak{p}$ and $\mathfrak{d}$ as different. $$\text{Tr}_{K_{\mathfrak{P}} / k_{\mathfrak{p}}} \mathcal{O}_{\mathfrak{P}} \subset \mathfrak{p} \iff \text{Tr}_{K_{\mathfrak{P}} / k_{\mathfrak{p}}} (\mathfrak{p}^{-1}\mathcal{O}_{\mathfrak{P}}) \subset \mathcal{O}_\mathfrak{P} \iff \mathfrak{p}^{-1}\subset \mathfrak{d}^{-1} \iff \mathfrak{d}\subset \mathfrak{p}=\mathfrak{P}^e$$ if and only $\mathfrak{P}^e \mid \mathfrak{d}$, which is equivalent to wild ramification.