$F \cong \mathbb Z_p$ for some prime p where $F$ is a field.

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....

Thank You!!

Let $\sigma$ be a ring homomorphism from $\mathbb Z$ to $F.$ Then what is the kernel of $\sigma?$
And what does the kernel of $\sigma$ tell us about $F?$
• yes... $p\mathbb Z$ will be the kernel for a field with char. p – user8795 Jan 5 '15 at 2:11
• And can the kernel of $\sigma$ be $(0)$? Namely, can $F$ have $0$ characteristic? – awllower Jan 5 '15 at 2:12
• So, is $\mathbb Z$ a field? – awllower Jan 5 '15 at 2:14