Ramification index ($e(L/K)$)

For a finite extension $L/K$ of non-Archimedean local fields, the ramification index $e(L/K)$ is the index of the value group of $K$ in that of $L$. Normalized valuations satisfy $v_L{|}_K=e(L/K)v_K$.

The residue-field degree of a finite extension of local fields is $f(L/K)=[k_L:k_K]$, and $[L:K]=e(L/K)f(L/K)$.

A finite extension of local fields is totally ramified when its residue-field degree is one, equivalently when its ramification index equals its field degree.