Definition 1

The index $[\mathfrak{p}_v : \Gamma_{v'}]$ is called the ramification index of $v'$ over $v$ (or over $\mathbf{K}$*) and denoted by* $e(v'/v)$ (or $e(A'/A)$, or sometimes $e(L/K)$*).

This index is a natural number or $+\infty$. If $v'_0$ is a valuation equivalent to $v'$, $e(v'/v)$ is also called the ramification index of $v'_0$ over $v$. If $e(v'/v) = 1$, $v'$ is called unramified over $v$.

On the other hand the residue field $\kappa(\mathbf{A})$ of $v$ is identified with a subfield of the residue field $\kappa(\mathbf{A}')$ of $v'$.