Endomorphism/Finite order/C/Characteristic polynomial/Exercise

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional ${}\mathbb {C}$-vector space ${}V$, and suppose that ${}\varphi$ has finite order. Show that the characteristic polynomial of ${}\varphi$ divides a polynomial of the form ${}{\left(X^{n_{1}}-1\right)}\cdots {\left(X^{n_{r}}-1\right)}$. Show also that the characteristic polynomial does not, in general, divide a polynomial of the form ${}X^{n}-1$.

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