Suppose $R$ is a complete local ring with unique maximal ideal $m$ and let $k$ be the residue field $R/m$. It is given that the characteristic of $k$ is zero. So it implies that $\mathrm{char}(R)$ is zero also. From here we can conclude two things.
$1.$ $R$ contains one copy of $\mathbb{Z}$.
$2.$ $\mathbb{Z}\cap m=0$.
But from here how can we conclude that
$1.$ $R$ contains a copy of $\mathbb{Q}$.
$2.$ $R$ contains a maximal subfield, say $L$, using Zorn's Lemma.
Actually I was reading the proof of Cohen's Structure Theorem, where all these steps come, but I'm stuck to prove the above two results.
Any help or hints would be highly appreciated.
Thanks in advance.