# Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?

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.

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NB: There is no continuous isomorphism. This illustrates the relevance of topological groups (as compared to abstract groups) in analytical contexts. –  Martin Brandenburg Feb 13 '13 at 22:59

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.
From what you saying, do you imply that there is a way to construct $\mathbb{R}$ without accepting AC? –  mez Feb 15 '13 at 8:39
And in this manner $\mathbb{R}\simeq\mathbb{R}/\mathbb{Q}$. –  Ash GX Nov 12 '13 at 15:18