Let $F$ be a field. Let $a \in F$ be a root of $f(x)$. The multiplicity of $a$ is the maximum positive integer $m$ such that $(x-a)^m |f(x)$.
I know that $x-a$ is an irreducible. How does this imply that the multiplicity of $a$ is well defined?
I know an argument where I can prove that the map $f(x)\mapsto (x-a) f(x)$ is injective. But I'd like to know the answer to the question above.
Essentially this boils down to proving that $\{m \in \mathbb{N} : (X- a)^m | f\}$ is bounded. If so, then its maximum is well defined. Suppose not: then, for each natural $m$ we have that $(X- a)^m | f$ and so there exists a polynomial $g_m$ such that
$$ g_m(X- a)^m = f. $$
If $f$ is non zero, then $g_m$ can't be zero for any $m$ and thus taking degrees we get that
$$ \deg f = \deg g_m + m \geq m \quad (\forall m \in \mathbb{N}) $$
which is absurd since $\deg f$ is a finite value.