Exercise 1, § 8

Let $K$ be a field and $A = K[[X, Y, Z]]$ the ring of formal power series in three indeterminates over $K$; let $v'$ (resp. $v''$) be the valuation on $A$ with values in the group $\mathbf{Z} \times \mathbf{Z}$ ordered lexicographically, such that $v'(X) = (1, 0)$, $v'(Y) = (0, 1)$, $v'$ is improper in $K[[Z]]$ (resp. $v''(X) = (1, 0)$, $v''$ is improper in $K[[Y]]$, $v''(Z) = (0, 1)$). Let $\sigma$ be the automorphism of $A$ leaving invariant $K$ and $X$ and such that $\sigma(Y) = Z, \sigma(Z) = Y$, if $B$ is the subring of $A$ consisting of the elements invariant under $\sigma$ and $E$ (resp. $F$) the field of fractions of $A$ (resp. $B$), the valuations $v'$ and $v''$ (extended canonically to $E$) have the same restriction $v$ to $F$, $F$ is complete with the topology defined by $v$ and $E$ is a quadratic extension of $F$; the two valuations $v', v''$ on $E$ are dependent but not equivalent.