Definition. root & multiplicity [1701]

Let $F$ be a field and $\overline{F}$ be an algebraic closure of $F$. Let $f(x) \in F[x]$. A root $\alpha \in \overline{F}$ of $f(x)$ is said to be of multiplicity $m$ if $$ (x - \alpha)^{m} \mid f(x), \text{ and } (x - \alpha)^{m+1} \nmid f(x) \text{ in } \overline{F}[x]. $$ A root of multiplicity $1$ is called a simple root. A root of multiplicity $\geq 2$ is called a multiple root.