In Wikipedia's article on the Brauer-Nesbitt theorem, they state that given a group $G$ and a field $E$, two *semisimple* representations $\rho_1,\rho_2 : G\longrightarrow \operatorname{GL}_n(E)$ are isomorphic if and only if $\rho_1(g),\rho_2(g)$ have the same characteristic polynomials for all $g\in G$.

The references they provide however only seem to apply to finite groups. Does anyone have a reference for the version stated above?

In characteristic 0, this appears in Lang's *Algebra* as Corollary 3.8 (p650)

In the general case, this is treated in Theorem 5.7 of Eggermont's masters thesis, but it feels sketchy to cite an unpublished masters thesis.

(EDIT: I forgot to include the stipulation that $\rho_1,\rho_2$ are semisimple)