Show that the prime subfield of a field of characteristic p is ringisomorphic to Zp and that the prime subfield of a field of characteristic 0 is ring-isomorphic to Q.

Let us first prove the following.
First Part: If F is a field of characteristic 0, then the prime subfield of F is isomorphic to Q.
Answer: Let us define a homomorphism

$\varphi :Z\to F\varphi :Z\to F$ by $\times \varphi (n)=n\times 1F.$
Clearly, the homomorphism $\varphi$ is injective, because F has characteristic 0. It follows that $\varphi (Z)$ is a subring of F and is isomorphic to Z.

Therefore, F contains a subfield isomorphic to Q. Since, Q has no subfields and so it is prime.

We know that, every field contains a unique prime subfield. Therefore, Q is the unique prime subfield of F.
Second Part: If F is a field of characteristic p,p, then the prime subfield of F is isomorphic to $Z_p$ .
Answer: Let us consider $\varphi :Z\to F$ as define in First Part. In this case, F has characteristic p, it follows that
Ker $\varphi =pZ$

Therefore, by the first isomorphism theorem that $\varphi (Z)$ is isomorphic to $\frac{Z}{pZ}.$

Clearly $\frac{Z}{pZ}.$ is a prime field. Since, every field contains a unique prime subfield. It follows that $\frac{Z}{pZ}$ is the unique prime subfield of F.