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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.

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  • $\begingroup$ If $V$ is the zero vector space, it is possible. $\endgroup$
    – user14972
    Commented Oct 4, 2017 at 23:01
  • $\begingroup$ Use the restrictive facts about cardinalities of vector spaces as constrained by the size of the base field. $\endgroup$
    – Randall
    Commented Oct 4, 2017 at 23:02
  • $\begingroup$ But what if V is not the zero vector? $\endgroup$ Commented Oct 4, 2017 at 23:04

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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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