Under the axiom of choice, $\mathbb{R}$ and $\mathbb{C}$ are both $2^{\aleph_0}$-dimensional vector spaces over $\mathbb{Q}$. Because they have the same dimension, they are isomorphic as vector spaces. That vector space isomorphism is also a group isomorphism between them.
The proof is clearly non-constructive. Is there a way to explicity construct such an isomorphism, or do we need the axiom of choice?
