I try to understand the following proof:
Let L/K be a finite, Galois extension. If L/K is unramified, then there is a canonical isomorphism Gal(L/K) $\cong$ Gal($k_{L}$/k) where $k_{L}$ is the residue field of L. Conversely, if Gal(L/K) $\cong$ Gal($k_{L}$/k), then L/K is ramified.
I try to understand the converse.
In the proof, the author uses the following expressions for the valution rings $O_{L}$ and O of L and K resp.
[L:K] = $[O_{L}:O]$ (I know $O_{L}$ is a free O-module of rank [L:K]. But what is $[O_{L}:O]$?
Then the author says: $[O_{L}:O]$ = $[O_{L}/(\pi):O/(\pi)]$ where $\pi$ is a prime element of K but we don't know whether it is prime or not in L. What is $[O_{L}/(\pi):O/(\pi)]$ and how do we know $[O_{L}:O]$ = $[O_{L}/(\pi):O/(\pi)]$?
Finally the author concludes $(\pi)$ is equal to $\mathfrak m_{L}$, so $\pi$ is a prime in L. Since I don't understand $[O_{L}:O]$ and $[O_{L}/(\pi):O/(\pi)]$, I can't follow why $[O_{L}/(\pi):O/(\pi)] = [O_{L}/\mathfrak m_{L}:O/(\pi)]$ implies $\mathfrak m_{L} = (\pi)$ in L.