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.

Question
Abstract algebra
asked 2021-02-09
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.

Answers (1)

2021-02-10
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.
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