 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.