Is there a change of basis that transforms any finite subgroup of $GL_n(C)$ into a subgroup of $GL_n(\bar{Q})$? I vaguely know that there is a related statement that is true... something like, if G is finite, then every representation of it can defined over some finite algebraic extension F of Q. (By defined over F, I mean that the representation is a map from G to $GL_n(F)$.) I'm not sure if this is the same as the question in my title. 
Can someone recommend a reference, or at least set the facts straight?
 A: Here's a somewhat indirect proof. It suffices to answer this question for irreducible representations. Consider the group algebra $\overline{\mathbb{Q}}[G]$, which is semisimple by Maschke's theorem. Hence by the Artin-Wedderburn theorem, it's a finite product of matrix algebras over finite-dimensional division algebras over $\overline{\mathbb{Q}}$. But since this field is algebraically closed, the only such division algebra is $\overline{\mathbb{Q}}$, so we conclude that
$$\overline{\mathbb{Q}}[G] \cong \prod_i M_{d_i}(\overline{\mathbb{Q}})$$
for some $d_i$. Tensoring further with $\mathbb{C}$ shows that the $d_i$ must be the dimensions of the irreducible representations of $G$, each of which occurs $d_i$ times in the decomposition of the above as a representation of $G$ in the factor $M_{d_i}$. Hence all of the irreducible representations of $G$ over $\mathbb{C}$ are defined over $\overline{\mathbb{Q}}$, and then the argument in the comments shows furthermore that they're defined over a finite extension $K$. 
Edit: Mariano's comment can be turned into a different answer with some work. Starting from a representation over $\mathbb{C}$ you get a representation over a finitely generated $\mathbb{Q}$-algebra. This has some maximal ideal, which you can quotient by to get a representation over a finitely generated field of characteristic $0$, which (by the Nullstellensatz) must be a finite extension of $\mathbb{Q}$. Then there's something to do to show that tensoring this up gives you what you started with. 
