Is it possible to define on $V = Z_{4}$ the structure of a vector space over $GF(2)$?

Question

Is it possible to define on $V = Z_{4}$ the structure of a vector space over $GF(2)$ in such a way that the vector addition is the usual addition in $Z_{4}$?

Answer 1

It is NOT.

$$ a+a = 0 $$

in linear spaces over GF(2) but not necessarily in $\mathbb Z_4,\ $ e.g. $1+1\ne 0$ in $\mathbb Z_4$.

This means that there is no isomorphism between the additive group of any linear space over GF(2) and the abelian group $\ \mathbb Z_4.$

Answer 2

The answer is no, and can be generalized. Let $K/GF(p^n)$ ($p$ is prime) be a vector space and let $a = a_1k_1 + \cdots + a_nk_n$ be an element in said vector space. Here, the $a_i$'s are elements in $GF(p^n)$ and the $k_i$'s are in $K$. See that adding $a$ to itself $p$ times will yield zero. (The characteristic of $GF(p^n)$ is $p$.) In fact, $ma = 0$ if and only if $m$ is a multiple of $p$.