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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.