Residue field degree of an extension of local fields (reviewed)

If $K$ is a finite extension of $\mathbb{Q}_p$, then let $\mathcal{O}_K$ denote its ring of integers. Let $P$ be its unique maximal ideal. Then $\mathcal{O}_K/P$ is a field which is a finite degree extension of $\mathbb{Z}_p/p\mathbb{Z}_p = \mathbb{F}_p$. The residue field degree is the degree of this extension: $f = [\mathcal{O}_K/P : \mathbb{F}_p]$.