MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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"...

share|cite|improve this question
up vote 5 down vote accepted

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.

share|cite|improve this answer
ah yes! thank you! – natura Jan 29 '10 at 7:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.