Exercise 19, § 21

Let $\pi$ be an irreducible complex representation of G of degree $>1$. Prove that there exists an element $x$ of G such that $\chi_{\pi}(x) = 0$. (Let $\{\lambda_1, . . . , \lambda_s\}$ be the orbit of $\pi$ in $\widehat{G}$ for the action of the group Gal($\mathbf{C}/\mathbf{Q}$), and let $\chi_1, . . . , \chi_s$ be the corresponding characters. Let $g\in G$; if $\chi (g)\not= 0$, then prove as in Exercise 17, c) that we have $|\chi_1(g). . . \chi_s(g)|\geqslant 1$, and therefore $\sum_i|\chi_i(g)|^2\geqslant s($FRV, III, §1, No. 1, p. 93, Proposition 2). If this holds for every $g\in G$, then deduce from the orthogonality relation for characters that we have equality, which leads to a contradiction by taking $g= 1.)*$