# $\mathbb R^n$ vector space over $\mathbb R$, $\mathbb F_p^n$ vector space over $\mathbb F_p$. Is $\mathbb R^n \cong \mathbb F_p^n$ as vector spaces?

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 $|\mathbb F_p^n|=p^n$.

• Two vector spaces over the same field and having the same dimension are isomorphic. – Lord Shark the Unknown Oct 18 '17 at 19:25
• Two vector spaces over the same field with same dimension are isomorphic. – Hagen von Eitzen Oct 18 '17 at 19:25
• In general, you cannot define a homomorphism between vector spaces over different fields, let alone an isomorphism. – Eclipse Sun Oct 18 '17 at 19:36

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.