Definition:Multiplicity (Polynomial)

Definition
Let $R$ be a ring.

Let $R \left[{X}\right]$ be the ring of polynomial forms over $R$.

Let $f \in R[X]$ be a nonzero polynomial.

An element $a\in R$ is said to be a root of multiplicity $n$ of $f$ if $n$ is the largest positive integer for which $(x-a)^n$ divides $f(x)$ in $R[x]$.

A double root is a root of multiplicity at least $2$.