Example (6)

Let Γ be a totally ordered group and k a field. Let Γ_+ be the monoid of positive elements of Γ and C the algebra of Γ_+ over k. By definition, C has a basis (x_\alpha)_{\alpha \in \Gamma_-} over k whose multiplication table is x_\alpha x_\beta = x_{\alpha + \beta}. If $x = \sum a_\alpha x_\alpha$ is a non-zero element of C, we write $v(x) = \inf_{a_\alpha \neq 0} (a)$ and $v(0) = +\infty$; it is immediately verified that the mapping $v$ of $C$ to $\Gamma_\infty$ satisfies conditions (VL_I) and (VL_{II}) of no. 1 and that $C$ is an integral domain. Let $K$ be the field of fractions of $C$ and $w$ the valuation on $K$ which extends $v$ (Proposition 4, no. 2). As every element of $\Gamma$ is the difference of two positive elements, $w$ admits $\Gamma$ as order group. Let $A$ be the ring of $w$ and $m$ its maximal ideal; we shall show that $A$ is the direct sum of $m$ and $k$ (identified with $k \cdot 1$), which will prove that the residue field of $w$ is isomorphic to $k$. Clearly $m \cap k = (0)$. On the other hand, denoting by $p$ the ideal of $C$ generated by the $x_a$ where $a > 0$, every element $x$ of valuation 0 in $K$ can be written in the form $(a + y)/(b + z)$ where $a \in k^*, b \in k^*, y \in p$ and $z \in p$; then

whence $w(x - ab^{-1}) > 0$ and $x \equiv ab^{-1} \pmod{m}$; this shows our assertion.

If $\Gamma = \mathbf{Z} \times \mathbf{Z}$, then $K = k(X, Y)$ and the above construction then provides valuations on $k(X, Y)$ which are improper on $k$, whose order group is $\mathbf{Z} \times \mathbf{Z}$ and whose residue field is $k$. These valuations depend on the order structure chosen on $\mathbf{Z} \times \mathbf{Z}$. For example, $\mathbf{Z} \times \mathbf{Z}$ can be given the lexicographic ordering. Or indeed, for an irrational number $a$, $\mathbf{Z} \times \mathbf{Z}$ may be identified with a subgroup of $\mathbf{R}$ under the homomorphism $(m, n) \mapsto m + n\alpha$ (a homomorphism which is injective since $a$ is irrational) and given the ordering induced by that on $\mathbf{R}$.

Other constructions of valuations using Proposition 4 of no. 2 will be described in § 10.