Eigenvalues are roots of cyclotomic polynomial

Question

I am reading Lyndon and Shupp's 'combinatorial group theory'. At page 25 it is stated that

if $g$ is an element of finite order $n$ in $\mathbb{GL}(2, \mathbb{Z})$, its eigenvalues must be roots of the cyclotomic polynomial $\Phi_n(x)$,$\ldots$

I can see why the eigenvalues are $n$th roots of unity, but I do not see why they should be primitive roots.

Answer 1

This is false. Take$$g=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$Then $g$ has order $2$, but only one of its eigenvalues is a root of $\Phi_2$.