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I am looking for a proof for why $\mathbb{C}$ and $\mathbb{R}$ are isomorphic under addition. I understand the outline of the proof; one shows that $\mathbb{R}$ forms a vector space over $\mathbb{Q}$, and $\mathbb{C}$ forms a vector space over $\mathbb{Q}$, and then show that both these have a basis with infinite dimension. However, I would like to see a rigorous proof. Does anyone have a good source for a proof of this?

Ty Jensen
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    What exactly is not rigorous in what you explained? – Captain Lama Mar 27 '20 at 16:53
  • @CaptainLama I can’t find a source for $\mathbb{C}$ forming a vector space over $\mathbb{Q}$ – Ty Jensen Mar 27 '20 at 16:55
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    It's not enough to say that they each have infinite dimension, you need that they each have the same infinite dimension (e.g. a countably-infinitely-dimensonal space over $\mathbb{Q}$ would not be isomorphic to $\mathbb{R}$). But that's easy to show (just look at the cardinalities of $\mathbb{R}$ and $\mathbb{C}$ themselves). – Noah Schweber Mar 27 '20 at 16:55
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    @TyJensen have you tried proving it? Is there somewhere you are stuck? Is that the only issue with the proof? It's hard to give a suitable answer without knowing where you are :). – Andres Mejia Mar 27 '20 at 16:57
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    @TyJensen "I can’t find a source for $\mathbb{C}$ forming a vector space over $\mathbb{Q}$." I doubt you'll find a source for that claim since it's basically trivial (if you want an overkill answer, every field is a vector space over each of its subfields). – Noah Schweber Mar 27 '20 at 16:57
  • I think that this question should be edited so that it includes the claims that @TyJensen is uncertain of, and it should be reopened in that case. – Andres Mejia Mar 27 '20 at 17:00
  • This likely is the most relevant duplicate (I edited the dupe listed to include it): Are $(\mathbb R, +)$ and $(\mathbb C, +)$ isomorphic as additive groups?. – amWhy Mar 27 '20 at 17:00
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    $\mathbb C$ forming a vector space over $\mathbb Q$. Try to prove it. Use the definition of "vector space". If you have trouble with that, then you will need to work on that before coming back to your original question. – GEdgar Mar 27 '20 at 17:00
  • @GEdgar So am I correct by assuming just showing that each axiom of a Vector space holds for any $a \in \mathbb{Q}$ and $z \in \mathbb{C}$ suffices to show its a vector space over $Q$? You are right, this is quite easy, but what about showing it has an infinitely dimensional basis (or the same dimension as the basis of $\mathbb{R}$ over $\mathbb{Q}$)? – Ty Jensen Mar 27 '20 at 17:08
  • There is more to do now. If $Z$ is an infinite-dimensional vector space over $\mathbb Q$, and $B$ is a Hamel basis of $Z$, then $|Z| = |B|$. – GEdgar Mar 27 '20 at 17:11

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