# Fields containing Z_p

1. Mar 2, 2008

### ehrenfest

[SOLVED] fields containing Z_p

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.

2. Mar 2, 2008

### morphism

The corollary is correct, but I don't quite understand what you're doing when you say things like "(p-1)+1 = 0, which is not true in Z_q". I mean, why is this even necessary to state?

3. Mar 2, 2008

### ehrenfest

If the field contains both Z_p and Z_q, where q>p, then the algebra of Z_p requires that (p-1) and 1 sum to 0. But the algebra of Z_q requires that (p-1) and 1 sum to p which is a contradiction.

Do you have a better way to prove the corollary?

4. Mar 2, 2008

### morphism

I guess I would just say, since p & q are distinct primes, p!=0 in Z_q. But it doesn't really matter, and what you did is fine.