# Valuation ring of completion of a field

I am actually quite confused. I have done an exercise that $$\mathbb{Z}_p$$ is a completion of $$\mathbb{Z}$$ w.r.t. the $$p$$-adic norm. Then again I got to know after reading somewhere that $$\mathbb{Z}_p$$ is also a completion of $$\mathbb{Z}_{(p)}$$. So is the completion of a valuation ring of field $$K$$ related to the valuation ring of the completion of $$K$$?

Another question I wanted to ask. Whether any complete field (where Cauchy sequences converge) is always a completion over another non complete field?

• The completion $\widehat{R}$ of a valuation ring $R$ at its valuation is a valuation ring. The valuation extends to the fraction field $Frac(R)$ and the valuation ring of $\widehat{Frac(R)}$ is the same as $\widehat{R}$. Note that it is not the same as the $\mathfrak{m}$-adic completion $\varprojlim R/\mathfrak{m}^n$. May 20 at 19:34
• @reuns So they are same.Now if I define the valuation ring of $(K,|\quad|)$ as the set $R=\{x\in K| |x|\leq1\}$ and that of completion $(\widehat{K},||\quad||)$ as the set $R^{\prime}=\{x\in K| ||x||\leq1\}$Then how can I show that $R^{\prime} = \widehat{R}$? Any hints or anything.I dont know actually what is m-adic completion. May 21 at 2:22