Roots multiplicity and square-free factorization of polynomials using companion matrices

Given an arbitrary monic polynomial $f$ over a field $F$ of characteristic 0, we use companion matrices to construct a polynomial $M_f\in F[X]$ of minimum degree such that for each root $\alpha$ of $f$ in the algebraic closure of $F$, $M_f(\alpha )$ is equal to the multiplicity $m(\alpha )$ of $\alpha$ as a root of $f$. As an application of $M_f$ we give a new method to compute in $F[X]$ each component of the square-free factorization $f=P_1{P}_2^2\cdots P_m^m$, where $P_k$ is the product of all $X-\alpha$ with $m(\alpha )=k$, for $k=1, \dots, m=\max m(\alpha)$.