I am given complete discrete valuation fields $K$ and $L$, finite extension $K\subset L$, corresponding d.v.r. $\mathcal O_K$ and $\mathcal O_L$ with maximal ideals $\mathfrak{m}_K$ and $\mathfrak{m}_L$. By $e$ I denote ramification index for $K\subset L$, that is $v_L(\pi_K)=e$ for $(\pi_K)=\mathfrak{m}_K.$ It is also noted that $\mathcal O_K/ \mathfrak{m}_K$ is finite.
I would like to learn how to prove $[L:K]=e[\mathcal O_L/\mathfrak{m}_L:\mathcal O_K/ \mathfrak{m}_K]$, however I did not quite understand proof from lecture. Does anyone know where can I find proof for that?
Thanks.