Exercise 28, § 21

Keep the assumptions of Exercise 27. Let $m$ be the l.c.m. of the orders of the $p$-regular elements of G. Let $F_0$ be the subfield of F generated by the $m$-th roots of unity; it is a finite field.

a) Prove that the character of any representation of G over F of finite degree takes all its values in $F_0$.

b) Prove that the quotient ring of $F_0[G]$ by its radical is a product of matrix algebras over $F_0$ (use a) and Wedderburn’s theorem). Deduce that the canonical homomorphism $R_{F_0}(G)\rightarrow R_F(G)$ is bijective.