Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?
I know that there is a bijection between $\mathbb{R}$ and $\mathbb{C}$, and this question asks whether they are isomorphic as abelian groups, are they referring to the additive abelian group? If so is there any simple isomorphism I can find? I know nothing about Hamel basis. Thanks.
Best Answer
Assuming the axiom of choice, yes.
Observe that both these abelian groups are actually $\mathbb Q$-vector spaces, and they have the same dimension, so they must be isomorphic as vector spaces, and such isomorphism is also a group isomorphism. This is in fact a stronger requirement than just group isomorphism, but nevermind that.
It is consistent with the failure of the axiom of choice that these two are not isomorphic, though. So one cannot give an explicit isomorphism between them.