Given a matrix $A\in \operatorname{SL}\_d(\mathbb{Z})$ ([the special linear group](https://en.wikipedia.org/wiki/Special_linear_group)) satisfying the two conditions: (1) no eigenvalue of $A$ is a root of unity, (2) the characteristic polynomial of $A$ is irreducible over $\mathbb{Q}$.



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> **QUESTION.** Does it follow that at least one eigenvalue $\lambda$ of $A$ fulfills $\vert\lambda\vert>1$? Assuming true, it seems that there must be a theorem of a sort here, but I couldn't recall. It would also be nice if one can relax the conditions to gain the same conclusion, if possible.
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**UPDATE.** After exploring papers by Kronecker (thanks [Terry Tao](https://mathoverflow.net/questions/415863/is-there-an-eigenvalue-of-modulus-larger-than-1#comment1066826_415863)) and others, I realized that we don't quite need "irreducibility" but "monic" is enough.



Is there an eigenvalue of modulus larger than 1?
Since the characteristic polynomial is irreducible, eigenvalues are simple and hence the matrix is $\mathbf{C}$-diagonalizable. If all were on the unit cercle, it would follow that $\{ A^n:n\in\mathbf{Z}\}$ is bounded. But since it is contained in the set of integral points, this would force it to be finite, and hence all eigenvalues would be roots of unity, contradiction.


Hence one is not on the unit circle. Since the product of eigenvalues is $\pm 1$ (the determinant), it follows that some eigenvalue has modulus $>1$.




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PS: Here's the straightforward way to extend this the general case (without irreducibility). Let $A\in\mathrm{M}\_d(\mathbf{Z})$ be a matrix whose eigenvalues are not only among $0$ and roots of unity. As any integral matrix, we can block-triangulate it (over $\mathbf{Z}$) so that all diagonal blocks are $\mathbf{Q}$-irreducible. Hence at least one diagonal block, with characteristic polynomial $P=P(t)$, also satisfies the assumption. So $P$ is $\mathbf{Q}$-irreducible, and not equal to $t$. If $|P(0)|\ge 2$ then clearly some root has modulus $>1$. Otherwise, $P(0)\in\{\pm 1\}$ and we are in the previous (main) case.


