Prove $\mathbb{Z}$ is not a vector space over a field
Solution 1:
The answer is, "Yes."
If $V$ is a vector space over a field of positive characteristic, then as an abelian group, every element of $V$ has finite order. If $V$ is a vector space over a field of characteristic $0$, then as an abelian group, $V$ is divisible. The abelian group $\mathbb Z$ has neither of these properties.