While defining Brauer character, we start with $\rho:G\rightarrow GL_n(F)$ an irreducible representation representation of group over a field $F$ of positive characteristic, and to each $p'$' element of $G$, we associate an algebraic integer in $\mathbb{C}$ (I hope one knows how this is done; see wiki); this function from $G_{p'}$ to $\mathbb{C}$ is Brauer character associated to irreducible representation of $G$ (over field of positive characteristic).

My question is: to study representation of $G$ over positive characteristic, why the values of Brauer characters are considered in $\mathbb{C}$? Couldn't we define the Brauer character corresponding to $\rho$ to be $trace(\rho(g))$? I mean why we should go in characteristic zero from positive characteristic?

I want to understand Brauer's intuition for this moving job from positive characteristic to zero characteristic. Almost every book defines Brauer charactewrs, but do not explains the "need" to associate values in $\mathbb{C}$ insted of the starting field $F$.


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