A set of notes I am reading claims the following:
For $L/K$ and $M/K$ extensions of $p$-adic fields, if $L/K$ is unramified then the natural map $\{K$-embeddings $L \hookrightarrow M\} \to \{k$-embeddings $k_L \hookrightarrow k_M\}$ is a bijection, where $k, k_L, k_M$ are the residue fields.
Could anyone explain why this is to me? It's completely glossed over and I don't find the claim entirely trivial. I can certainly see what the natural map is, but not necessarily why it's a bijection in an unramified extension $L/K$. If anything in my question is unclear please just ask and I'll clear it up. Many thanks.