Remark 1

If $f = 0$ we agree to say that $a$ has order $\geq h$ relative to $f$, whatever $a \in \mathbf{A}$ and the integer $h \geq 0$. For any $f \in \mathbf{A}[X]$ and $a \in \mathbf{A}$, to say that $a$ has order $\geq h$ relative to $f$ means that $(X - \alpha)^h$ divides $f$.