Idea: from number theory.
Take field extension $L/k$ and write $G = \Gal(L\slice k)$. 
Define the [inertia subgroup](inertia%20group) $I\leq G$.
Then $L$  is

- Ramified iff $I = \emptyset$.
	- Equivalently, $\OO_L/\mfm_{\OO_k} \OO_L$ is a field
- Totally unramified iff $I = G$