added new tag (characteristic p)
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of Z/pZ or of Zp is the same for odd primes, but not for 2. Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for 2. So in these examples characteristic 2 is a messy special case.
On the other hand, certain types of combinatorial questions can be reduced to linear algebra over F2, and this relationship doesn't seem to generalize to other finite fields. So in this example characteristic 2 is a nice special case.
Is anything deep going on here? (I have a vague idea here about additive inverses and Fourier analysis over Z/2Z, but I'll wait to see what other people say.)