Existence and uniqueness of unramified local extensions

For a non-Archimedean local field $K$, every finite extension $\ell /k_K$ of its residue field is realized by an unramified extension of $K$, unique up to $K$-isomorphism. Lift the irreducible polynomial of a residue generator to a monic integral polynomial and adjoin a root. Its residue degree equals its field degree, forcing ramification index one. In any other such extension the Hensel lemma lifts the same simple residue root, giving an isomorphism.