Lemma 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 $\Gamma'$. There exists a unique valuation w on $K(X)$ such that, for $P = \sum_j a_j X^j$ ($a_j \in K$), $w(P) = \inf_j (v(a_j) + j\xi)$.

By Proposition 4 of § 3, no. 2, it suffices to show that the formula

defines a valuation on the ring $K[X]$. As

it follows that

for $P, Q$ in $K[X]$, equality holding if $w(P) \neq w(Q)$. We show that

for $P = \sum a_j X^j$ and $Q = \sum b_j X^j$. Let $i$ (resp. $k$) be the least of the integers $j$ such that $v(a_j) + j\xi$ (resp. $v(b_j) + j\xi$) attains its minimum; let $a$ (resp. $\beta$) denote this minimum. For $j, j'$ in $\mathbf{N}$,

whence $w(PQ) \geq a + \beta$ by (2). Consider now the term $c X^{i+k}$ of degree $i + k$ in $PQ$; then $c = \sum_{n \in \mathbf{Z}} a_{i+n} b_{k-n}$; by the choice of $i$ and $k$, the element

takes its minimum value $a + \beta$ once and once only with $n = 0$; hence $w(c X^{i+k}) = a + \beta$, whence, by (1),