Example

Let $k$ be a field,

$(X, Y$ indeterminates). Let $B = E[Y]_{(Y)}$ be the valuation ring of $K$ defined by the extremal element $Y$ of the principal ideal domain $E[Y]$ (§ 1, no. 4, Proposition 3). The residue field $\kappa(B)$ is canonically identified with $E[Y]/(Y) = E$. Similarly, let $A' = k[X]_{(X)}$ be the valuation ring of $E = k(X)$ defined by the extremal element $X$ of $k[X]$. Denoting by $h$, the place of $E$ associated with $B$ and writing $A = h_B^{-1}(A')$, a valuation ring $A$ of $K$ is defined which is contained in $B$ and $\kappa(A) = \kappa(A') = k$. The canonical place $h$: $K \to k$ can be described as follows: iff $(X, Y)$ is an element of $K$, then we first put $Y = 0$ inf (which gives an element of $\tilde{E} = k(X)^*$), then $X = 0$ in the result obtained. The groups $\Gamma_{A'}$ and $\Gamma_B$ are canonically isomorphic to $\mathbf{Z}$ (§ 3, no. 4, Example 4). It is not difficult to show (cf. § 10, no. 2, Lemma 2) that the group $\Gamma_A$ is isomorphic to the lexicographic product $\mathbf{Z} \times \mathbf{Z}$ and that the valuation $v_A$ is equivalent to the valuation defined in § 3, no. 4, end of Example 6.