Proposition 1

Let $K$ be a field, $v$ a valuation on $K$, $\Gamma$ its order group, $\Gamma'$ a totally ordered group containing $\Gamma$ and $\xi$ an element of $\Gamma'$ such that the relations $n\xi \in \Gamma, n \in \mathbf{Z}$ imply $n = 0$. Then there exists a unique valuation w on $K(X)$ with values in $\Gamma'$ and extending $v$ such that $w(X) = \xi$. The residue field of w is equal to that of $v$ and its order group is the subgroup $\Gamma + \mathbf{Z}\xi$ of $\Gamma'$.

We show first the uniqueness of w. Let $P = \sum_j a_j X^j$ be an element of $K[X]$. Then $w(a_j X^j) = v(a_j) + j\xi$, which shows that the monomials $a_j X^j$ such that $a_j \neq 0$ have distinct values for w. It follows that $w(P) = \inf_j (v(a_j) + j\xi)$, which shows both the uniqueness of w on $K[X]$ (hence also on $K(X)$) and the fact that the order group of w is $\Gamma + \mathbf{Z}\xi$. It is further seen that, if $P \neq 0$, we may write $P = a X^n (1 + u)$, where $a \in K^*, n \in \mathbf{N}, u \in K(X)$ and $w(u) > 0$; every element $R \neq 0$ of $K(X)$ can therefore be written in the form

where $b \in K^*, n \in \mathbf{Z}, u' \in K(X)$ and $w(u') > 0$; then $w(R) = v(b) + n \xi$, hence $w(R) = 0$ if and only if $v(b) = 0$ and $n = 0$; thus, when $w(R) = 0$, $R$ and $b$ are congruent modulo the ideal of $w$, which shows that the residue field of $w$ is equal to that of $v$.

Finally the existence of $w$ follows from Lemma 1.