Question

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"...

Answer 1

The coefficients of the characteristic polynomial of $A^n$ are symmetric functions of the roots of the characteristic polynomial of $A$, so the result follows from the Fundamental Theorem of Symmetric Functions.