To reinforce Geoff's answer, I'd emphasize that the "fairly standard exercise" mentioned in the question is only standard when you consider characteristic 0 irreducible representations as is usually done in a first course (then it's also usual to start with a big field like $\mathbb{C}$ to avoid extra complications). When dealing with modular representations, there are lots of further issues.
Qiaochu attempts an outline of the "standard exercise" which shows clearly where problems arise. In his first step, you need characteristic 0 (and a splitting field) to be confident that the degree of an irreducible representation divides the group order. This is definitely false in general. And then of course characteristic 2 is especially dangerous for dealing with the negative of the $2 \times 2$ identity matrix.
The bottom line is that you have to avoid this "standard exercise" and focus just on the special situation you are studying, in order to avoid confusion.