Can the real field be made a vector space over the complex field?

Question

We all know that $ \mathbb{C} $ is naturally a vector space over $ \mathbb{R} $. However, is there some kind of (possibly weird) scalar multiplication law that would make $ \mathbb{R} $ a vector space over $ \mathbb{C} $ instead?

Answer 1

Yes. As an additive group, $\mathbb{R}$ is isomorphic to $\mathbb{C}$ (you can see this, for example, from the fact that they're both continuum-dimensional vector spaces over $\mathbb{Q}$). Since the additive group $\mathbb{C}$ can be made into a complex vector space, so can the isomorphic group $\mathbb{R}$.

Answer 2

Well, note that as abelian groups $(\mathbb C,+)$ and $(\mathbb R,+)$ are isomorphic. Since a vector space is merely an abelian group with scalar multiplication you can just pick a homomorphism between $\mathbb R$ and $\mathbb C$ as additive groups, and use that to define a vector space.

In fact you can do the same trick with any finitely dimensional space over $\mathbb C$.