number of elements on a finite vector space I know that there is no vector space over any field $\mathbb{F}$ having precisely $6$ elements. I would like to know if there is a theorem which characterizes those $n's \in \mathbb{N}$ such that there is a vector space having exactly $n$ elements over a field $\mathbb{F}$. For example: $n=p^{k}$ where $p$ is a prime number and $k\in \mathbb{N}$ ($k\neq 0$).

Let V be a finite vector space over a field F.
Since any two distinct scalars multiplied to a non zero vector will give distinct vectors,and since V has only finite number of elements, F must be finite. Hence the number of elements of F must be the n-th (say) power of some prime p.
Since V is finite, it must be m- dimensional for some m. So V is isomorphic to the direct sum of m copies F. So the number of elements of V must be the mn-th power of p.
Therefore, the number of elements of a non zero finite vector space also must have the number of elements a power of a prime number.
