Let $C$ be a finite-dimensional field extension of the real numbers, so $(C \setminus 0)/{\mathbb R}_+$
is a compact abelian Lie group, and a sphere of dimension $\dim_{\mathbb R} C - 1$. If $C \neq {\mathbb R}$ so this group is connected, then it's a $K(\pi,1)$ (proved using the exponential map, which is a surjective group homomorphism). In particular, it can't be a sphere of dimension $\geq 2$, so $\dim C = 2$.
(Anybody know who this proof is due to? I believe I heard Mazur or Gross, but I'm not sure.)