Chapter 2: pg. 89 problem 5
If F has characteristic 2, then since (v+w) + (u+w) = u + v + 2w = u + v we have Q3 collinear with Q1 and Q2 [Note that the computation only uses commutativity of addition and the fact that 2 = 0 in a division ring of characteristic 2]. On the other hand, suppose that Q3 is on the line Q1Q2. Then there exist a,b in F so that u + v = a(v+w) + b(u+w) which can be rewritten as 0 = (b-1)u + (a-1)v + (a+b)w. Since u,v and w are a basis, all the coefficients must be 0. Thus, a = 1, b = 1 and a + b = 1 + 1 = 0 and the division ring F has characteristic 2.