Exercise 17, § 21

Let C be a conjugacy class of G and $\pi$ an irreducible representation of G of degree $d$.

a) Prove that the element $d^{-1}$ Card(C) $\chi (C)$ of K is integral over $\mathbf{Z}$ (reason as in the proof of Proposition 9 of VIII, p. 415).

b) Suppose that $d$ and Card(C) are mutually prime integers. Prove that the element $d^{-1}\chi (C)$ is integral over $\mathbf{Z}$.

$*$c) Suppose, moreover, $K =\mathbf{C}$. Prove that all conjugates of $d^{-1}\chi (C)$ over $\mathbf{Q}$ have absolute value $\leqslant 1$. Deduce from Exercise 16, c) that $|\chi (C)|$ is zero or equal to $d$. Conclude that if $\chi (C)$ is not zero, then $\pi (x)$ is a homothety for every $x\in C$. In particular, if $\pi$ is faithful and C is not reduced to one element, then we have $\chi (C) = 0.*$