Suppose $\mathbb{F}$ is a field. Define $\pi:\mathbb{Z}\to \mathbb{F}$ so that $\pi(0)=0$ and $\pi(n+1)=\pi(n)+1$ for all $n \geq 0$. If $n<0$ then $\pi(n)=-\pi(-n)$. So $\pi$ is a homomorphism.
Suppose $\pi$ is not $1-1$. Then I have shown the smallest $n$ such that $\pi(n)=0$ is prime and $\operatorname{char}\mathbb{F}=n$
Can anyone help me prove that $\operatorname{im}(\pi)$ is a subfield of $\mathbb{F}$. It's easy to show it's an additive subgroup, but how do you show that it contains inverses for all it's elements?
Also how is this the smallest subfield and what is $|\operatorname{im}(\pi)|$ as it must be finite.