Proposition 4

— Let $f$ and $g$ be non-zero elements of $\mathbf{A}[X]$. Let $a \in \mathbf{A}$, and let the orders of $a$ relative to $f$ and $g$ be $p$ and $q$ respectively. (i) The order of $a$ relative to $f + g$ is $\geq \inf(p, q)$. It is equal to $\inf(p, q)$ if $p \neq q$. (ii) The order of $a$ relative to $fg$ is $\geq p + q$. It is equal to $p + q$ if $\mathbf{A}$ is an integral domain.

For we have $f(X) = (X - \alpha)^p f_1(X)$, $g(X) = (X - \alpha)^q g_1(X)$ with $f_1(\alpha) \neq 0$, $g_1(a) \neq 0$. Suppose for example that $p \leq q$; then we have

and if $p < q$, $a$ is not a root of $f_1(X) + (X - \alpha)^{q-p} g_1(X)$; this proves (i). On the other hand, we have $f(X)g(X) = (X - \alpha)^{p+q} f_1(X)g_1(X)$ and $f_1(\alpha)g_1(\alpha) \neq 0$ if $A$ is an integral domain; this proves (ii).