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$(\Bbb{R},+) $ and $(\Bbb{C}, +) $ are group-isomorphic.

Consider, two vector space $\Bbb{R}_{\Bbb{Q}}$ and $\Bbb{C}_{\Bbb{Q}}$.

Any two vector space over the same field having same dimension are linearly isomorphic.

Suppose, $T: \Bbb{R}_{\Bbb{Q}}\to \Bbb{C}_{\Bbb{Q}}$ be an Invertible linear map.

And the additive structure are group isomorphic.

Question :

  1. Is the above way is correct?

  2. I know that $\Bbb{R}_{\Bbb{Q}}$ and $\Bbb{C}_{\Bbb{Q}}$ are infinite dimensional vector space but how to find the cardinality of their hamel basis or how to prove their respective basis have the same cardinality?

Sourav Ghosh
  • 12,997
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    No, this is not enough. You need to show your claim. Your proof should more look like here or here at the duplicates. – Dietrich Burde Mar 14 '22 at 09:15
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    Since C is obviously isomorphic to R x R as additive groups, it's enough to argue that R is infinite dimensional over Q and that the product of an infinite cardinal M with itself is M. – Ned Mar 14 '22 at 11:52

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