习题 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 $K[[Y]]$, $v''(Z) = (0, 1)$). 令$\sigma$为$A$的自同构，留下不变$K$和$X$，并且使得$\sigma(Y) = Z, \sigma(Z) = Y$，如果$B$是$A$由……组成的子环，不变元素$\sigma$和$E$（分别$F$）域或$A$（分别$B$）的分数，估值$v'$和$v''$（规范地扩展到$E$）具有相同的限制$v$到$F$，$F$与$v$定义的拓扑完整，$E$是$F$的二次扩张； $v', v''$ 对 $E$ 的两个估值是相关的，但不相等。