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1. The problem statement, all variables and given/known data

Here is a theorem in my book: "A field F is either of prime characteristic p and contains a subfield isomorphic to Z_p or of characteristic 0 and contains a subfield isomorphic to Q."

2. Relevant equations

3. The attempt at a solution

Here is my corollary: "If a field contains a copy of Z_p, then it must be of prime characteristic p."

Here is the proof: If the field has prime characteristic q not equal to p, then it must contain a copy of Z_q. If q > p, then since the field contains Z_p, we have (p-1)+1 = 0, which is not true in Z_q. If the q < p, we have (q-1) + 1 = 0, which is not true in Z_p. Therefore, the field cannot contain Z_q where q is not equal to p. Furthermore, if the field contains Q, then (p-1)+1=0 is a contradiction. Therefore, the only possibility that the above theorem gives is that the characteristic is p.

Please confirm that this is correct.

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# Homework Help: Fields containing Z_p

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