Remark

The valuations $v_f$ and $\bar{w}$ introduced in the proof of Proposition 13 are the essential valuations of $A[X]$. It will be sufficient for us to show that, if $V$ is the set of valuations $v_f$ ($f$ irreducible) and $\bar{w}$ (w essential), then, for all $v' \in V$, there exists an element $g \in K(X)$ which is not in $A[X]$ and such that $v''(g) \geq 0$ for all the valuations $v'' \in V$ distinct from $v'$; this will prove that $V - \{v'\}$ does not satisfy (AK$_{II}$) and the conclusion will follow therefore from no. 4, Corollary 2 to Proposition 6. Suppose first that $v'$ is of the form $\bar{w}$: then we may take $g$ to be an element $b \in K$ such that $w(b) < 0, w'(b) \geq 0$ for the essential valuations $w'$ of $A$ distinct from $w$, for then $v_f(b) = 0$ for every irreducible monic polynomial $f$ in $K[X]$; the existence of an element $b$ satisfying the above conditions follows from no. 5, Proposition 9. Suppose secondly that $v'$ is of the form $v_f$ for an irreducible monic polynomial $f \in K[X]$ of degree $m$; then we may take $g = a/f$ where $a \in A$. For $v_h(g) \geq 0$ for every irreducible monic polynomial $h \neq f$ in $K[X]$; it remains to choose $a \in A$ such that, for every essential valuation $w$ of $A$, $w(a)$ is at least equal to the greatest lower bound of the elements $w(c_i)$, where the $c_i$ are the coefficients of $f$ ($1 \leq i \leq m$); now the existence of such an $a \in A$ follows from (AK$_{III}$) and no. 5, Proposition 9.

We may also say (no. 6, Theorem 4) that the prime ideals of $A[X]$ of height 1 are: (1) the prime ideals of the form $pA[X]$, where $p$ is a prime ideal of $A$ of height 1; (2) the prime ideals of the form $m \cap A[X]$, where $m$ is a (necessarily principal) prime ideal of $K[X]$.

The latter are characterized by the fact that their intersection with $A$ is reduced to 0.