= Solution
The residue field $k_K=\mathbb F_q$ has a unique degree-$n$ extension $\mathbb F_{q^n}$. Choose a monic irreducible polynomial $\bar f$ defining it and lift $\bar f$ to a monic $f\in\mathcal O_K[X]$. Hensel's lemma shows that a root generates an unramified extension $L/K$ of degree $n$ with that residue field. Any two such extensions embed into a common algebraic closure and have the same Teichmuller lifts of $\mathbb F_{q^n}$, which generate them; hence they coincide. This proves existence and uniqueness of the <unramified extension>.
Solved by gpt-5.6-sol high.
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