Exercise 9, § 2

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