# Vector space with new scalar multiplication defined over finite field still vector space?

If $V$ is a vector space (not the zero vector space) over $\mathbb{R}$, and if $F$ is a finite field. how could I show that it is not possible to define a new scalar multiplication of $F$ on $V$, in a way that $V$ with this scalar multiplication and the usual addition becomes a vector space over $F$?

• If $V$ is the zero vector space, it is possible. – user14972 Oct 4 '17 at 23:01
Let the characteristic of $F$ be $p$. Then:
$$\vec 0 = 0 \cdot \vec v = \underbrace{(1+1+1+\cdots+1)}_{p\text{ terms}} \vec v \ne \underbrace{\vec v + \vec v + \vec v + \cdots + \vec v}_{p\text{ terms}} = p\vec v$$