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# characteristic of residue field and characteristic of the ring

I am looking for a proof of the result that appears at the end of page 418 here indicating the possibilities for equal characteristic and mixed characteristic.

I could prove the following: Let $$(R,m,K=R/m)$$ be a local ring.

1) If char $$K=0$$, then, char $$R=0$$. So $$R$$ contains a copy of $$\mathbb{Z}$$. Now, since $$R/m$$ has char $$0$$, none of the images of non-zero integers in $$R$$ can be in $$m$$, else their images in $$R/m$$ will be zero contradicting char $$K=0$$. Then by the universal property of localization, $$R$$ contains $$\mathbb{Q}$$.

2) If char $$K=p$$, then either char $$R$$ is $$0$$ or non-zero (I don't understand why it should be a prime power if it's non-zero. If char $$R=p>0$$ where $$p$$ is a prime, then, $$R$$ contains a copy of $$\mathbb{Z}/p\mathbb{Z}$$.

So I guess the proof now reduces to proving the question I raise in the paranthesis and why $$R$$ cannot contain a field in mixed characteristic. Any help will be appreciated.

 asked Jun 30 '11 at 3:55 Brittany Murphy 3122 bronze badges