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Let $\mathbb R^n$ be vector space over $\mathbb R$ and let $\mathbb F_p^n$ be a vector space over $\mathbb F_p$.

Since any two vector spaces with equal dimension are isomorphic, does this mean that $\mathbb R^n \cong \mathbb F_p^n$?

I can't see how this is so since $|\mathbb F_p^n|=p^n$.

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    $\begingroup$ Two vector spaces over the same field and having the same dimension are isomorphic. $\endgroup$ – Lord Shark the Unknown Oct 18 '17 at 19:25
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    $\begingroup$ Two vector spaces over the same field with same dimension are isomorphic. $\endgroup$ – Hagen von Eitzen Oct 18 '17 at 19:25
  • $\begingroup$ In general, you cannot define a homomorphism between vector spaces over different fields, let alone an isomorphism. $\endgroup$ – Eclipse Sun Oct 18 '17 at 19:36
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No. $\mathbb{F}_p^n$ has $p^n$ elements, while $\mathbb{R}^n$ has uncountably many elements. That is, there is no set bijection between these two.

The statement you're claiming needs to be refined: any two vector spaces over the same field $K$ of the same dimension are isomorphic.

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