If a Noetherian local ring contains a field, then so does its $\mathfrak m$-adic completion?

Question

Let $(R, \mathfrak m)$ be a Noetherian local ring containing a field. Then, does the $\mathfrak m$-adic completion $\widehat R$ of $R$ also contain a field?

Answer 1

Let $R$ be a local ring and $K$ a subfield. Then $1_K(1_K-1_R)=0$ so one of $1_K,1_K-1_R$ is in $\mathfrak{m}$, the other won't be as $(1_K,1_K-1_R)=R$, so one is a unit, either $1_K=0$ or $1_K-1_R=0$, of course it must be the latter.

Now let $R$ be any ring and $I\subsetneq R$ any ideal and $K$ a subfield such that $1_K=1_R$.

The natural map $f:R\to \widehat{R}=\varprojlim R/I^n$ is a ring homomorphism, $\widehat{R}$ is not the zero ring, and $f(1_K)=1_{\widehat{R}}$ so $f(K)$ is a subfield of $\widehat{R}$.