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Given an element $\gamma$ in $GL(n,F)$, where $F$ is either a global field or a non archimedean local field.

Assume $\gamma$ is elliptic, i.e. its characteristic polynomial irreducible. Let $Z(F)$ be the center of $GL(n,F)$.

Is $\gamma$ conjugated to an element in $Z(F)GL(n, o)$, where $o$ is the ring of integers of $F$?

Heuristic: The centralizer of $\gamma$ is compact modulo the centrum.

What about $GL(n ,\mathbb{R})$ and $GL(n, \mathbb{C})$, where $GL(n,o)$ is replaced by $O(n)$ respective $U(n)$?

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For the highlighted question, take a scalar multiple of the matrix to have entries in the ring of integers if necessary, and assume that the matrix is integral with irreducible characteristic polynomial. Then its characteristic polynomial is (up to sign) monic with integral coefficients. It is conjugate to the companion matrix for the characteristic polynomial, which has integral entries. I'm not sure what conditions are being imposed for the follow-up question.

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    Yes, that wasn't quite the mistake I made- I was (I think!) just thinking that the companion matrix was invertible, which need not be the case of course. Eg you could work with ${\rm GL}(2,\mathbb{Z})$ and the companion matrix for $x^{2} - 2.$2012-03-15