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 ϕ:Z\rightarrow Fϕ:Z\rightarrow F by \times \phi (n)=n\times 1F.

Clearly, the homomorphism \phi is injective, because F has characteristic 0. It follows that \phi (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 Zp .

Answer: Let us consider \phi :Z\rightarrow F as define in First Part. In this case, F has characteristic p, it follows that

Ker \phi =pZ

Therefore, by the first isomorphism theorem that \phi (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.

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 ϕ:Z\rightarrow Fϕ:Z\rightarrow F by \times \phi (n)=n\times 1F.

Clearly, the homomorphism \phi is injective, because F has characteristic 0. It follows that \phi (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 Zp .

Answer: Let us consider \phi :Z\rightarrow F as define in First Part. In this case, F has characteristic p, it follows that

Ker \phi =pZ

Therefore, by the first isomorphism theorem that \phi (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.