= Lifting residue-field embeddings of unramified extensions
For an <unramified extension> $L/K$ of non-Archimedean <local fields> and any finite extension $M/K$, reduction is a bijection between $K$-<field homomorphisms> $L\to M$ and $k_K$-<field homomorphisms> $k_L\to k_M$. Choose an integral generator with separable irreducible reduction. Each specified residue image lifts uniquely to a root in $M$ by <Hensel lemma>. Conversely a $K$-embedding preserves the unique extended <valuation> and therefore induces the residue embedding. The same argument applies when $M$ is any non-Archimedean local extension with the given compatible embedding of $K$.
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