Corollary 1

Let K be a field, v a valuation on K, L a quasi-Galois extension of K and $v'$ and $v''$ two extensions of v to L. Then there exists a K-automorphism s of L such that $v''$ is equivalent to $v' \circ s$.

Let $f'$ and $f''$ be the places of K associated with $v'$ and $v''$; replacing them if need be by equivalent places, it may be assumed that they both take their values in the algebraic closure of the residue field of v (no. 1, Proposition 1). Then there exists a K-automorphism s of L such that $f'' = f' \circ s$ (Proposition 7); thus $v''$ is equivalent to $v' \circ s$ by virtue of the correspondence between places and valuations (§ 3, no. 3).