Lemma 2

— Let N be an A-module, n an integer $\geq 0$ and

Suppose that there exist $\alpha_0, \alpha_1, \ldots, \alpha_r \in A$ such that $f(\alpha_0) = \cdots = f(\alpha_r) = 0$, and such that for $i \neq j$ the homothety of ratio $\alpha_i - \alpha_j$ in N is injective, then $f = 0$.

(This lemma generalizes the Cor. of IV, p. 16.)

The lemma clearly holds for $n = 0$; we shall prove it by induction on $n$. We have

where g is an element of N[X] of the form $m_0' + m_1' X + \ldots + m_{n-1}' X^{n-1}$. The hypotheses of the lemma imply that $g(\alpha_1) = \cdots = g(\alpha_r) = 0$, hence $g = 0$ by the induction hypothesis, and so $f = 0$.