Exercise 9, § 2

Let $k$ be an integer $> 0$ such that $k.1 = 0$ in $A$. Let $h$ be an integer $> 0$ and let $f$ be the polynomial $(X - \alpha)^{k+h} + (X - \alpha)^k$. Then $\alpha$ is a root of order $k$ of $f$, and a root of order $\geq k + h - 1$ of $Df$.