Let $G$ be any group and $k$ be any field. While studying representation theory, I saw some theorems which are special cases of the following theorem:
$\textbf{Theorem. }$ Let $\rho_1,\rho_2:G\to \text{GL}_n(k)$ be two semisimple representations of $G$ over $k$. If characteristic polynomial of $\rho_1(g)$ and $\rho_2(g)$ are the same for all $g\in G$, then $\rho_1\simeq \rho_2$.
$\textbf{Question}$: Is this theorem true?
Since I couldn't find proof of this theorem with full generality, I tried to prove it myself. After a bit of work, I could prove the theorem if $G$ is finite or $k$ is perfect, and I posted it here. Now I wonder whether this theorem is true or false if $G$ is infinite and $k$ is imperfect. I suspect there is a counterexample since an important lemma in my proof (Theorem 4 of the link) is no longer true. However, I cannot find any counterexample yet.
Adding an answer so this is not unanswered.
The theorem is true, more generally for finite-dimensional semisimple representations of any $k$-algebra $A$. Short proof is given in an answer by David Speyer here.
The key observation is that you can reduce to the case where $A$ is a finite-dimensional semisimple $k$-algebra.