Ramification index of a prime in a local field (reviewed)

If $F$ is a finite extension of $\Q_p$ and $K$ a finite extension of $F$. Then $\mathcal{O}_F$ and $\mathcal{O}_K$, the ring of integers of $F$ and $K$ are discrete valuation domains, so they have unique maximal ideals $P_F$ and $P_K$ which are principal. If $P_F=(\pi_F)$, the element $\pi_F$ is a uniformizer for $F$.

The principal ideal $\pi_F\mathcal{O}_K=P_K^e$ for some positive integer $e$. The integer $e$ is the ramification index for $K$ over $F$. The ramification index of $K$ is then the ramification index for $K$ over $\Q_p$.

If $e=1$, then we say that the extension is unramified, and if $e=[K:\Q_p]$, then we say that the extension is totally ramified.