Show that Z cannot be turned into a vector space over any field.

Question

Show that Z cannot be turned into a vector space over any field.

So, we have 2 cases here.

Case 1:lets suppose the charF=P,

n does not equal 0, then (1+1+...+1)n=1n+1n+...+1n=n+n+...+n=pn=wchich does not equal zero

=0n

=0 ................................................ a contradiction

Case 2: CharF=0 ->Q

?????

Answer 1

If $\Bbb Z$ were a vector space over some field $F$, then it must be a 1-dimensional vector space, since $\Bbb Z$ is generated as an additive group by 1. Now simply note that $\Bbb Z$ is not a field.