Proposition 9

— Let $u \in A[(X_i)_{i \in I}]$; then for u to be nilpotent in the ring $A[(X_i)_{i \in I}]$ it is necessary and sufficient for all its coefficients to be nilpotent in the ring A.

As in the proof of Prop. 8 we can make a reduction to the case of polynomials in one variable X. If all the coefficients of $u$ are nilpotent, then $u$ is nilpotent (I, p. 99, Cor. 1). Suppose that $u$ is nilpotent but not zero and let $n$ be its degree; we shall argue by induction on $n$. Let $a$ be the leading coefficient of $u$. There exists an integer $m > 0$ such that $u^m = 0$. The leading coefficient of $u^m$ is $a^m$, hence $a^m = 0$. Now $u - \alpha X^n$ is nilpotent (I, loc. cit.) and the induction hypothesis shows all the coefficients of $u - \alpha X^n$ to be nilpotent. Thus all the coefficients of $u$ are nilpotent.