Recently, I have been studying about Character Theory of Finite Groups, mostly from "Groups and Representations" by J. Alperin & R. Bell. In the aforementioned textbook, the characters of a finite group $G$ is studied using the the group algebra of $G$ over the field of complex numbers. However, I know the characters of a finite group $G$ could be studied over any field $F$.
So, these are my questions, for which not only do I appreciate your answers, but, more importantly, I would be grateful if you could mention good references for supplementary studies, of course referring to the adequate parts which answer my following questions:
Let assume we want to study characters of a finite group $G$ over a field $F$. Would it be possible to obtain all the information we need to know about the irreducible characters of $G$ over $F$ by studying the characters of $G$ over appropriate subfields (or overfields) of $F$. If so, how could one make the vague idea of "appropriate subfield (or overfield)" a bit more precise?
We define the virtual characters (sometimes referred to as generalized characters) of a group $G$ as all the $\mathbb Z$-linear combinations of the irreducible characters of $G$, which forms a ring. What special information do we obtain by this generalization?
Thanks for your help in advance. In particular, good references, with precise address for reading more about these questions would be highly appreciated.