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

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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