How to show two infinite-dimensional vector spaces are not isomorphic

Solution 1:

Two vector spaces (over the same field) are isomorphic iff they have the same dimension - even if that dimension is infinite.

Actually, in the high-dimensional case it's even simpler: if $V, W$ are infinite-dimensional vector spaces over a field $F$ with $\mathrm{dim}(V), \mathrm{dim}(W)\ge \vert F\vert$, then $V\cong W$ iff $\vert V\vert=\vert W\vert$. In particular, if $F=\mathbb{Q}$, two infinite-dimensional vector spaces over $F$ are isomorphic iff they have the same cardinality.

So, for example:

• As vector spaces over $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ are isomorphic (this assumes the axiom of choice).

• The algebraic numbers $\overline{\mathbb{Q}}$ are not isomorphic to the complex numbers $\mathbb{C}$ as vector spaces over $\mathbb{Q}$, since the former is countable while the latter is uncountable.

EDIT: All of this assumes the axiom of choice - without which, the idea of "dimension" doesn't really make sense. See the comments for a bit more about this.