Definition 15.123.1. We say that $A \to B$ or $A \subset B$ is an extension of valuation rings if $A$ and $B$ are valuation rings and $A \to B$ is injective and local. Such an extension induces a commutative diagram
\[ \xymatrix{ A \setminus \{ 0\} \ar[r] \ar[d]_ v & B \setminus \{ 0\} \ar[d]^ v \\ \Gamma _ A \ar[r] & \Gamma _ B } \]
where $\Gamma _ A$ and $\Gamma _ B$ are the value groups. We say that $B$ is weakly unramified over $A$ if the lower horizontal arrow is a bijection. If the extension of residue fields $\kappa _ A = A/\mathfrak m_ A \subset \kappa _ B = B/\mathfrak m_ B$ is finite, then we set $f = [\kappa _ B : \kappa _ A]$ and we call it the residual degree or residue degree of the extension $A \subset B$.
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