Let $F$ be a field. If char$(F) = p>0$, show that the prime subring of $F$ is isomorphic to the field $\mathbb{F}_{p}$, and if char$(F) = 0$, then the prime subring is isomorphic to $\mathbb{Z}$.
I have shown that if char$(F) = p>0$ then $p$ is prime. Now, I tried to build a homomorphism $\varphi: \mathbb{F}_{p} \longrightarrow P$ but I couldn't conclude the bijective. Thanks for any hint!
The natural map $\pi:\mathbb Z\to F$ given by $1\mapsto 1_F$ and extended additively is automatically onto the prime subring of $F$ in any case. If this is your difficulty, then maybe I can spend more time talking about it. Just ask.
Then the first homomrphism theorem says that $\mathbb Z/\ker(\pi)$ is isomorphic to its image, the prime subring.
Since the image is a domain, $\ker(\pi)$ is a prime ideal in $\mathbb Z$, and we know what the options are for those.