The minimal polynomials of powers of cycles in the ordinary representations of symmetric and alternating groups

Denote the alternating and symmetric groups of degree $n$ by $A_n$ and $S_n$ respectively. Consider a permutation $\sigma \in S_n$ all of whose nontrivial cycles are of the same length. We find the minimal polynomials of $\sigma$ in the ordinary irreducible representations of $A_n$ and $S_n$.