Isomorphisms between $\Bbb R^n$ and fields

Question

Are there real vector spaces with dimension $\geq 3$ that are isomorphic to a field? I case $n=2$ there are the complex numbers and for $n=3$ the quaternions are non-commutative.

Thanks in advance.

Answer 1

A vector space cannot be isomorphic to a field (they are two different types of structures) but you could ask how many finite dimensional $\Bbb R$ algebras there are which are division rings or fields.

It turns out there are only three possibilities for division rings: $\Bbb R$, $\Bbb C$ and $\Bbb H$. The case of $\Bbb H$ corresponds to $n=4$, not $3$. There is nothing for $3$, and nothing above $4$. If you want the $\Bbb R$ algebra to be a field, there are only the two possibilities $\Bbb R$ and $\Bbb C$.

The result that says $\Bbb R$, $\Bbb C$ and $\Bbb H$ are the only finite dimensional $\Bbb R$-division algebras is the Frobenius theorem. There are many extensions of this theorem. One such extension allows the division algebra to be nonassociative, and it scoops up the $8$-dimensional octonions in addition to these three. (But it doesn't get any more than that!)