# Characteristic of a field in a vector space

Given a vector space $$V$$ over a field $$F$$, if $$F$$ has (nonzero) characteristic $$n$$, then for any $$v\in V$$,

\begin{align*} v+\cdots+v \textrm{ (n times)} &= (1_F+\cdots+1_F \textrm{ (n times)})\cdot v \\ &= 0_F\cdot v \\ &= 0 \end{align*}

That is, the vector space appears to "inherit" the characteristic of its field.

This seems to allow us to rule out the possibility of constructing a vector space using certain sets. For instance, even though $$\mathbb{Z}_5\times\mathbb{Z}_2$$ is an abelian group, it can't be a vector space over any field, since $$(0,1)+(0,1)=(0,0)$$ but $$(1,0)+(1,0)\neq(0,0)$$.

Is this correct? If so, is there a name for this idea? Is it related to any ideas with deeper significance?

• Any finite-dimensional vector space over a finite field has a number of elements which is the power of a prime. Indeed, let the ground field $\mathbb K$ have $p^r$ elements (for some prime $p$). Then the vector space is isomorphic to $\mathbb K^d$ where $d$ is the dimension, which has $p^{rd}$ elements Nov 11, 2022 at 13:37

a finite abelian group $$G$$ carry a vector space structure over a finite field $$\mathbb{F}$$ iff the order of any nontrivial element has order $$Char(\mathbb{F})$$, in which case it is isomorphic to $$\mathbb{F}^{n}$$, in particular a necessary condition is that $$\lvert G \rvert =Char(\mathbb{F})^{n}$$,So any abelian group whose order is not a power of some prime cannot admit a vector space structure over any finite field.