Given a field $K$ (for example $K = \mathbb Q$) and a polynomial $p \in K[X]$ with cyclic Galois group $C_n$ then one only needs to adjoin an $n$th root of some element $k \in K(\zeta_n)$ to $K(\zeta_n)$ to get the splitting field $L=K(\zeta_n,\sqrt[n]{k})$ of $p$. It is possible to actually find the number $k$.
It is said that the roots of $p$ lie in this field $L$, I would like to know how you can find these roots?