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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"...

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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.

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ah yes! thank you! –  natura Jan 29 '10 at 7:37

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