 Solution

Make $F^n$ an $F[X]$-module by $Xv=Av$. It is finitely generated and torsion, so the structure theorem gives cyclic summands $F[X]/(d_i)$ with $d_1\mid \cdots \mid d_k$. In the power basis of each summand, multiplication by $X$ has the companion matrix of $d_i$. Their block diagonal sum is the rational canonical form, unique because the invariant factors are unique.