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In the general case, this is treated in Theorem 5.7 of Eggermont's masters thesis, but it feels sketchy to cite an unpublished masters thesis.

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In the general case, this is treated in Theorem 5.7 of Eggermont's masters thesis, but it feels sketchy to cite an unpublished masters thesis.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

In Wikipedia's article on the Brauer-Nesbitt theorem, they state that given a group $G$ and a field $E$, two 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?

This is treated in Eggermont's masters thesis, but it feels sketchy to cite an unpublished masters thesis.

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.

In Wikipedia's article on the Brauer-Nesbitt theorem, they state that given a group $G$ and a field $E$, two representations $\rho_1,\rho_2 : G\longrightarrow GL_n(E)$ 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?

This is treated in Eggermont's masters thesis, but it feels sketchy to cite an unpublished masters thesis.

In Wikipedia's article on the Brauer-Nesbitt theorem, they state that given a group $G$ and a field $E$, two 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?

This is treated in Eggermont's masters thesis, but it feels sketchy to cite an unpublished masters thesis.