Exercise 5, § 4

Let $A$ be the subring of $\mathbf{R}$ generated by the numbers $2.\cos(\pi/m(s, s'))$. Show that $A$ is a free $\mathbf{Z}$-module of finite type, and that the matrices of the $\sigma(w)$ for $w \in W$ have their coefficients in $A$. Deduce that the coefficients of the characteristic polynomials of the $\sigma(w)$ are algebraic integers.