linear algebra - an exercise on integrality of characteristic polynomials

Suppose A is a matrix with coefficient in $Q_{ell}$, and all the coefficients of its char. polynomial are in $Z$ (thus an integral polynomial). Prove that the char. polynomial of $A^n$ is also integral. (This question probably has nothing to do with the base field　$Q_{ell}$)

This question actually comes from a remark in Serre's book "abelian l-adic representations", so allow me to tag it with "number theory"...