I'm reading the proof of the Deligne-Serre theorem attaching Galois representations to newforms of weight one, and there's a representation-theoretic argument that I don't understand at all.
The setup is as follows: there is a finite field $F$ of characteristic $\ell$ and a two-dimensional semisimple representation $\rho:G \rightarrow GL_2(F)$ (where $G$ is a finite group). We want to show that it can be realized over $\mathbb{F}_{\ell}$.
The argument seems as follows: the Brauer group of a finite field is trivial, so it is enough to show that the representation is isomorphic to its conjugates. But for each element of $G$, its image in $GL_2(F)$ has a characteristic polynomial in $\mathbb{F}_{\ell}$, so it works.
This suggests several questions:
1) If $\rho,\rho'$ are two representations of the same finite group $|G|$ with the same characteristic polynomial, are they isomorphic?
It holds in characteristic zero because the representations are direct sums of irreducible representations and the characters of irreducible representations are linearly independent functions. But does the argument hold in prime characteristic, at least if $\rho,\rho'$ are semisimple and two-dimensional?
(I don't have much access to libraries, so I'd appreciate it if you could briefly recall the argument or the main steps -- everything I know about representations is over $\mathbb{C}$ where all these subtleties do not occur)
2) Let $\rho$ be a semisimple representation over some Galois extension $K$ of a field $k$ that is isomorphic to all its conjugates. In general, $\rho$ isn't isomorphic to a representation defined over $k$ (If a representation is Galois-invariant, is it defined over the smaller field?). So why does this hold when $k$ is a finite field, and what is the relationship with the Brauer group?
Thank you.