Proposition 1

Suppose that K is a commutative field. Let $\pi$ and $\pi '$ be finite-dimensional semisimple K-linear representations of G.

a) If the characteristic polynomials of $\pi (g)$ and $\pi '(g)$ are the same for every $g\in G$, then the representations $\pi$ and $\pi '$ are isomorphic.

b) Suppose, moreover, that K has characteristic 0. If the characters $\chi_{\pi}$ and $\chi_{\pi'}$ are equal, then the representations $\pi$ and $\pi '$ are isomorphic.

Assertion a) follows from Corollary 1 of VIII, p. 386; assertion b) follows from part a) of the corollary of VIII, p. 384.