Square matrix with rational coefficients having $k$-th root

The general result is as follows:

Let $A\in M_n(\mathbb{Q})$; then

For every integer $k$, there exists $B\in M_n(\mathbb{Q})$ such that $B^k=A$ IFF $A(A-I_n)^n=0$.

You can see several proofs

in english in https://artofproblemsolving.com/community/c7h42444 , grobber's post.

in french in "La revue de mathématiques spéciales" n° 117-2 , R535, where, with a colleague, we gave a solution using Mahler's measure.