TITLE: Vector space with new scalar multiplication defined over finite field still vector space?
QUESTION [1 upvotes]: 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 $F$?
Thanks in advance.

REPLY [2 votes]: Let the characteristic of $F$ be $p$. Then:
$$\vec 0 = 0 \cdot \vec v = \underbrace{(1+1+1+\cdots+1)}_{p\text{ terms}} \vec v \ne \underbrace{\vec v + \vec v + \vec v + \cdots + \vec v}_{p\text{ terms}} = p\vec v$$
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