Solving polynomials by radicals with roots of unity in minimum depth

Let $k$ be an algebraic number field. Let $\alpha$ be a root of a polynomial $f\in k[x]$ which is solvable by radicals. Let $L$ be the splitting field of $\alpha$ over $k$. Let $n$ be a natural number divisible by the discriminant of the maximal abelian subextension of $L$, as well as the exponent of $G(L/k)$, the Galois group of $L$ over $k$. We show that an optimal nested radical with roots of unity for $\alpha$ can be effectively constructed from the derived series of the solvable Galois group of $L(\zeta _n )$ over $k(\zeta _n )$.