Proposition 5

Let $K$ be a field, v a valuation on $K$, $A$ its ring, m its ideal, L a finite extension of $K$ of degree $n$, $B$ the integral closure of $A$ in $L$ and $(v'_i)_{1 \leq i \leq s}$ a complete system of extensions of $v$ to $L$. Then

Let $A_i$ be the ring of $v'_i$; then $A_i = B_{m_i}$ where $m_i$ runs through the family of maximal ideals of $B$ (no. 3, Remark). Let $q_i$ be the saturation of $mB$ with respect to $m_i$ (Chapter II, § 2, no. 4). By Chapter V, Corollary 3 to Proposition 1, no. 1, § 2, the canonical homomorphism $B/mB \to \prod_{i=1}^s B/q_i$ is an isomorphism and $m_i$ is the only maximal ideal of B containing $q_i$. Hence $B/q_i$ is canonically isomorphic to $(B/q_i)_{m_i}$ (Chapter II, § 3, no. 3, Proposition 8), that is to

Therefore there is a canonical isomorphism $B/mB \to \prod_{i=1}^s A_i/mA_i$, whence the result by virtue of Proposition 4 of no. 4.