Proposition 7

Let K be a field, L a quasi-Galois extension of K and f and $f'$ places of L with values in the same field F. Suppose that the restrictions off and $f'$ to K coincide. Then there exists a K-automorphism s of L such that $f' = f \circ s$.

Let A be the ring of the place of K the common restriction off and $f'$. The rings off and $f'$ contain the integral closure $A'$ of A in L (§ 1, no. 3, Corollary 3 to Theorem 3) and hence (Chapter V, § 2, no. 3, Corollary 1 to Proposition 6) there exists a K-automorphism s of L such that the restrictions off and $f \circ s$ to $A'$ are equal; if $m'$ is the common kernel of these restrictions, $m' \cap A$ is the maximal ideal of A, hence $m'$ is a maximal ideal of $A'$ and the places $f'$ and $f \circ s$ coincide on the ring $A'_{m'}$; but by Proposition 6 the only valuation ring of L dominating $A'_{m'}$ is the ring $A'_{m'}$ itself and hence the rings of the places $f'$ and $f \circ s$ are the same.