Unramified extensions of local fields

Question

Let $(K,\nu)$ be a local field. Let $L,K’$ be two unramified extensions of $K$ such that $$[L:K]=[K’:K]=n$$ then $L \cong K’$. Searching on web I found a prof on “algebraic number theory” of Milne (page 127-128), but it hasn’t more details. Let $k$ be the residue field of $K$ and let $k’$ be the residue field of $L$ and $K’$. $K’L$ is unramified over $K$ since $K’$ and $L$ are unramified over $K$ (I found this prof on neukirch book page 153-154) with residue field $k’$, so $$[LK’:K]=[k’:k]=[K’:K] $$ and this prove that $L\cong K’$. I do not understand because the residue field of $LK’$ is $k’$, can anyone explane this?