Bourbaki

Corollary

Tag 04WQ, permanent. Cite it as [Bourbaki, Tag 04WQ].

Commutative Algebra, chapter VI, § 8. Extensions of a valuation to an algebraic extension

*With the same hypotheses and notation,

[B/mB:A/m]=∑i=1sε(vi′/v)f(vi′/v)≤∑i=1sε(vi′/v)f(vi′/v)≤n. [B/mB : A/m] = \sum_{i=1}^s \varepsilon(v'_i/v) f(v'_i/v) \leq \sum_{i=1}^s \varepsilon(v'_i/v) f(v'_i/v) \leq n.
We know that ε(vi′/v)≤ε(vi′/v)\varepsilon(v'_i/v) \leq \varepsilon(v'_i/v) (no. 4, Corollary to Proposition 3) and ∑i=1sε(vi′/v)f(vi′/v)≤n\sum_{i=1}^s \varepsilon(v'_i/v) f(v'_i/v) \leq n (no. 3, Theorem 1).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Transcribed from the printed volume. Not translated.