# 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.

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

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Derrick

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

$$\phi:Z\rightarrow F\phi:Z\rightarrow F$$ 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 $$Z_{p}$$ .
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.