Here we have $F$ a field and $f: \mathbb{Z} \to F$ a ring epimorphism. We are to prove that $F$ is a finite field with non-zero characteristic.
I know that since $f$ is an epimorphism, we have that $\forall x\in F$, $\exists y \in \mathbb{Z}$ such that $f(x) = y$. (As an epimorphism is a surjective ring homomorphism).
I am unsure about two things: Am I to prove that F is finite, or just prove that it has a non-zero characteristic. Either way, I could use some guidance. Thank you :)
PS This is for HW, so I only require a good hint about how to approach the problem.
Observe that $F$ is isomorphic to $\mathbb Z/ \ker f$.
On your particular request: since you are asked to show that $F$ is a finite field you must show this. The characteristic is then always finite. The converse is not true though. Consider the quotient field of the polynomial ring over a finite field as example.