Suppose $F$ is a field and there is a ring homomorphism from $\mathbb Z$ onto $F$. Show that $F \cong \mathbb Z_p$ for some prime $p$.
My try: If $F$ is of finite characteristics then it prime. So by first isomorphism theorem $F \cong \mathbb Z/p\mathbb Z$ and we know that $\mathbb Z/p\mathbb Z \cong \mathbb Z_p$ and hence the proof and on the other hand if $F$ is not of finite characteristics then its characteristics is zero and hence....
Is it correct explanation?? Please Help!!